Linear Algebra Examples

$(- 2 x z) (- 2 y z) [\begin{array}{c} - 2 x^{2} & - 2 x y & - 2 x z \\ - 2 x y & 1 - 2 y^{2} & - 2 y z \\ - 2 x z & - 2 y z & 1 - 2 z^{2} \end{array}]$

Step 1

Multiply $z$ by $z$ by adding the exponents.

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Step 1.1

Move $z$ .

$- 2 x (z \cdot z) (- 2 y) [\begin{array}{c} - 2 x^{2} & - 2 x y & - 2 x z \\ - 2 x y & 1 - 2 y^{2} & - 2 y z \\ - 2 x z & - 2 y z & 1 - 2 z^{2} \end{array}]$

Step 1.2

Multiply $z$ by $z$ .

$- 2 x z^{2} (- 2 y) [\begin{array}{c} - 2 x^{2} & - 2 x y & - 2 x z \\ - 2 x y & 1 - 2 y^{2} & - 2 y z \\ - 2 x z & - 2 y z & 1 - 2 z^{2} \end{array}]$

Step 2

Multiply $- 2$ by $- 2$ .

$4 x z^{2} y [\begin{array}{c} - 2 x^{2} & - 2 x y & - 2 x z \\ - 2 x y & 1 - 2 y^{2} & - 2 y z \\ - 2 x z & - 2 y z & 1 - 2 z^{2} \end{array}]$

Step 3

Multiply $4 x z^{2} y$ by each element of the matrix.

$[\begin{array}{c} 4 x z^{2} y (- 2 x^{2}) & 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4

Simplify each element in the matrix.

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Step 4.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.1.1

Move $x^{2}$ .

$[\begin{array}{c} 4 (x^{2} x) z^{2} y \cdot - 2 & 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.1.2

Multiply $x^{2}$ by $x$ .

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Step 4.1.2.1

Raise $x$ to the power of $1$ .

$[\begin{array}{c} 4 (x^{2} x^{1}) z^{2} y \cdot - 2 & 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} 4 x^{2 + 1} z^{2} y \cdot - 2 & 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.1.3

Add $2$ and $1$ .

$[\begin{array}{c} 4 x^{3} z^{2} y \cdot - 2 & 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.2

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.3

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1

Move $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & 4 (x \cdot x) z^{2} y (- 2 y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.3.2

Multiply $x$ by $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & 4 x^{2} z^{2} y (- 2 y) & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.4

Multiply $y$ by $y$ by adding the exponents.

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Step 4.4.1

Move $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & 4 x^{2} z^{2} (y \cdot y) \cdot - 2 & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.4.2

Multiply $y$ by $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & 4 x^{2} z^{2} y^{2} \cdot - 2 & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.5

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y (- 2 x z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.6

Multiply $x$ by $x$ by adding the exponents.

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Step 4.6.1

Move $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 (x \cdot x) z^{2} y (- 2 z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.6.2

Multiply $x$ by $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 x^{2} z^{2} y (- 2 z) \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.7

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 4.7.1

Move $z$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 x^{2} (z \cdot z^{2}) y \cdot - 2 \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.7.2

Multiply $z$ by $z^{2}$ .

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Step 4.7.2.1

Raise $z$ to the power of $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 x^{2} (z^{1} z^{2}) y \cdot - 2 \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 x^{2} z^{1 + 2} y \cdot - 2 \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.7.3

Add $1$ and $2$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & 4 x^{2} z^{3} y \cdot - 2 \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.8

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ 4 x z^{2} y (- 2 x y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.9

Multiply $x$ by $x$ by adding the exponents.

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Step 4.9.1

Move $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ 4 (x \cdot x) z^{2} y (- 2 y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.9.2

Multiply $x$ by $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ 4 x^{2} z^{2} y (- 2 y) & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.10

Multiply $y$ by $y$ by adding the exponents.

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Step 4.10.1

Move $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ 4 x^{2} z^{2} (y \cdot y) \cdot - 2 & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.10.2

Multiply $y$ by $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ 4 x^{2} z^{2} y^{2} \cdot - 2 & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.11

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y (1 - 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.12

Apply the distributive property.

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y \cdot 1 + 4 x z^{2} y (- 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.13

Multiply $4$ by $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y + 4 x z^{2} y (- 2 y^{2}) & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.14

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 4.14.1

Move $y^{2}$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y + 4 x z^{2} (y^{2} y) \cdot - 2 & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.14.2

Multiply $y^{2}$ by $y$ .

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Step 4.14.2.1

Raise $y$ to the power of $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y + 4 x z^{2} (y^{2} y^{1}) \cdot - 2 & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.14.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y + 4 x z^{2} y^{2 + 1} \cdot - 2 & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.14.3

Add $2$ and $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y + 4 x z^{2} y^{3} \cdot - 2 & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.15

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x z^{2} y (- 2 y z) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.16

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 4.16.1

Move $z$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x (z \cdot z^{2}) y (- 2 y) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.16.2

Multiply $z$ by $z^{2}$ .

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Step 4.16.2.1

Raise $z$ to the power of $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x (z^{1} z^{2}) y (- 2 y) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.16.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x z^{1 + 2} y (- 2 y) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.16.3

Add $1$ and $2$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x z^{3} y (- 2 y) \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.17

Multiply $y$ by $y$ by adding the exponents.

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Step 4.17.1

Move $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x z^{3} (y \cdot y) \cdot - 2 \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.17.2

Multiply $y$ by $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & 4 x z^{3} y^{2} \cdot - 2 \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.18

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 x z^{2} y (- 2 x z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.19

Multiply $x$ by $x$ by adding the exponents.

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Step 4.19.1

Move $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 (x \cdot x) z^{2} y (- 2 z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.19.2

Multiply $x$ by $x$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 x^{2} z^{2} y (- 2 z) & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.20

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 4.20.1

Move $z$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 x^{2} (z \cdot z^{2}) y \cdot - 2 & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.20.2

Multiply $z$ by $z^{2}$ .

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Step 4.20.2.1

Raise $z$ to the power of $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 x^{2} (z^{1} z^{2}) y \cdot - 2 & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.20.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 x^{2} z^{1 + 2} y \cdot - 2 & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.20.3

Add $1$ and $2$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ 4 x^{2} z^{3} y \cdot - 2 & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.21

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y (- 2 y z) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.22

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 4.22.1

Move $z$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x (z \cdot z^{2}) y (- 2 y) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.22.2

Multiply $z$ by $z^{2}$ .

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Step 4.22.2.1

Raise $z$ to the power of $1$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x (z^{1} z^{2}) y (- 2 y) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.22.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{1 + 2} y (- 2 y) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.22.3

Add $1$ and $2$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{3} y (- 2 y) & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.23

Multiply $y$ by $y$ by adding the exponents.

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Step 4.23.1

Move $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{3} (y \cdot y) \cdot - 2 & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.23.2

Multiply $y$ by $y$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{3} y^{2} \cdot - 2 & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.24

Multiply $- 2$ by $4$ .

$[\begin{array}{c} - 8 x^{3} z^{2} y & - 8 x^{2} z^{2} y^{2} & - 8 x^{2} z^{3} y \\ - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} & 4 x z^{2} y (1 - 2 z^{2}) \end{array}]$

Step 4.25

Apply the distributive property.

Step 4.26

Multiply $4$ by $1$ .

Step 4.27

Multiply $z^{2}$ by $z^{2}$ by adding the exponents.

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Step 4.27.1

Move $z^{2}$ .

Step 4.27.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 4.27.3

Add $2$ and $2$ .

Step 4.28

Multiply $- 2$ by $4$ .

Step 5

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 5.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x z^{3} y^{2} & 4 x z^{2} y - 8 x z^{4} y \end{array} |$

Step 5.4

Multiply element $a_{11}$ by its cofactor.

$- 8 x^{3} z^{2} y | \begin{array}{c} 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x z^{3} y^{2} & 4 x z^{2} y - 8 x z^{4} y \end{array} |$

Step 5.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} |$

Step 5.6

Multiply element $a_{12}$ by its cofactor.

$8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} |$

Step 5.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 5.8

Multiply element $a_{13}$ by its cofactor.

$- 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 5.9

Add the terms together.

$- 8 x^{3} z^{2} y | \begin{array}{c} 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x z^{3} y^{2} & 4 x z^{2} y - 8 x z^{4} y \end{array} | + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 4 x z^{2} y - 8 x z^{2} y^{3} & - 8 x z^{3} y^{2} \\ - 8 x z^{3} y^{2} & 4 x z^{2} y - 8 x z^{4} y \end{array} |$ .

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Step 6.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 8 x^{3} z^{2} y ((4 x z^{2} y - 8 x z^{2} y^{3}) (4 x z^{2} y - 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2

Simplify the determinant.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Expand $(4 x z^{2} y - 8 x z^{2} y^{3}) (4 x z^{2} y - 8 x z^{4} y)$ using the FOIL Method.

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Step 6.2.1.1.1

Apply the distributive property.

$- 8 x^{3} z^{2} y (4 x z^{2} y (4 x z^{2} y - 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y - 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.1.2

Apply the distributive property.

$- 8 x^{3} z^{2} y (4 x z^{2} y (4 x z^{2} y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y - 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.1.3

Apply the distributive property.

$- 8 x^{3} z^{2} y (4 x z^{2} y (4 x z^{2} y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2

Simplify each term.

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Step 6.2.1.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1.2.1.1

Move $x$ .

$- 8 x^{3} z^{2} y (4 (x \cdot x) z^{2} y (4 z^{2} y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.1.2

Multiply $x$ by $x$ .

$- 8 x^{3} z^{2} y (4 x^{2} z^{2} y (4 z^{2} y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.2

Multiply $z^{2}$ by $z^{2}$ by adding the exponents.

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Step 6.2.1.2.2.1

Move $z^{2}$ .

$- 8 x^{3} z^{2} y (4 x^{2} (z^{2} z^{2}) y (4 y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (4 x^{2} z^{2 + 2} y (4 y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.2.3

Add $2$ and $2$ .

$- 8 x^{3} z^{2} y (4 x^{2} z^{4} y (4 y) + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.3

Multiply $y$ by $y$ by adding the exponents.

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Step 6.2.1.2.3.1

Move $y$ .

$- 8 x^{3} z^{2} y (4 x^{2} z^{4} (y \cdot y) \cdot 4 + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.3.2

Multiply $y$ by $y$ .

$- 8 x^{3} z^{2} y (4 x^{2} z^{4} y^{2} \cdot 4 + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.4

Multiply $4$ by $4$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x z^{2} y (- 8 x z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1.2.5.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 (x \cdot x) z^{2} y (- 8 z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.5.2

Multiply $x$ by $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x^{2} z^{2} y (- 8 z^{4} y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.6

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 6.2.1.2.6.1

Move $z^{4}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x^{2} (z^{4} z^{2}) y (- 8 y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x^{2} z^{4 + 2} y (- 8 y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.6.3

Add $4$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x^{2} z^{6} y (- 8 y) - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.7

Multiply $y$ by $y$ by adding the exponents.

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Step 6.2.1.2.7.1

Move $y$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x^{2} z^{6} (y \cdot y) \cdot - 8 - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.7.2

Multiply $y$ by $y$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} + 4 x^{2} z^{6} y^{2} \cdot - 8 - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.8

Multiply $- 8$ by $4$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x z^{2} y^{3} (4 x z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.9

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1.2.9.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 (x \cdot x) z^{2} y^{3} (4 z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.9.2

Multiply $x$ by $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{2} y^{3} (4 z^{2} y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.10

Multiply $z^{2}$ by $z^{2}$ by adding the exponents.

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Step 6.2.1.2.10.1

Move $z^{2}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} (z^{2} z^{2}) y^{3} (4 y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.10.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{2 + 2} y^{3} (4 y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.10.3

Add $2$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{4} y^{3} (4 y) - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.11

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 6.2.1.2.11.1

Move $y$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{4} (y \cdot y^{3}) \cdot 4 - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.11.2

Multiply $y$ by $y^{3}$ .

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Step 6.2.1.2.11.2.1

Raise $y$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{4} (y^{1} y^{3}) \cdot 4 - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{4} y^{1 + 3} \cdot 4 - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.11.3

Add $1$ and $3$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 8 x^{2} z^{4} y^{4} \cdot 4 - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.12

Multiply $4$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x z^{2} y^{3} (- 8 x z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.13

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1.2.13.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 (x \cdot x) z^{2} y^{3} (- 8 z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.13.2

Multiply $x$ by $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{2} y^{3} (- 8 z^{4} y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.14

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 6.2.1.2.14.1

Move $z^{4}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} (z^{4} z^{2}) y^{3} (- 8 y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.14.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{4 + 2} y^{3} (- 8 y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.14.3

Add $4$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{6} y^{3} (- 8 y) - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.15

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 6.2.1.2.15.1

Move $y$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{6} (y \cdot y^{3}) \cdot - 8 - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.15.2

Multiply $y$ by $y^{3}$ .

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Step 6.2.1.2.15.2.1

Raise $y$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{6} (y^{1} y^{3}) \cdot - 8 - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{6} y^{1 + 3} \cdot - 8 - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.15.3

Add $1$ and $3$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} - 8 x^{2} z^{6} y^{4} \cdot - 8 - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.2.16

Multiply $- 8$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x z^{3} y^{2} (- 8 x z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1.3.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 (x \cdot x) z^{3} y^{2} (- 8 z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.3.2

Multiply $x$ by $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} z^{3} y^{2} (- 8 z^{3} y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.4

Multiply $z^{3}$ by $z^{3}$ by adding the exponents.

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Step 6.2.1.4.1

Move $z^{3}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} (z^{3} z^{3}) y^{2} (- 8 y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} z^{3 + 3} y^{2} (- 8 y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.4.3

Add $3$ and $3$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} z^{6} y^{2} (- 8 y^{2}))) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.5

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 6.2.1.5.1

Move $y^{2}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} z^{6} (y^{2} y^{2}) \cdot - 8)) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} z^{6} y^{2 + 2} \cdot - 8)) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.5.3

Add $2$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (- 8 x^{2} z^{6} y^{4} \cdot - 8)) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.6

Multiply $- 8$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - (64 x^{2} z^{6} y^{4})) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.1.7

Multiply $64$ by $- 1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - 64 x^{2} z^{6} y^{4}) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.2

Combine the opposite terms in $16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 64 x^{2} z^{6} y^{4} - 64 x^{2} z^{6} y^{4}$ .

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Step 6.2.2.1

Subtract $64 x^{2} z^{6} y^{4}$ from $64 x^{2} z^{6} y^{4}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4} + 0) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 6.2.2.2

Add $16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}$ and $0$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} | - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7

Evaluate $| \begin{array}{c} - 8 x^{2} z^{2} y^{2} & - 8 x z^{3} y^{2} \\ - 8 x^{2} z^{3} y & 4 x z^{2} y - 8 x z^{4} y \end{array} |$ .

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Step 7.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{2} z^{2} y^{2} (4 x z^{2} y - 8 x z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2

Simplify the determinant.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Apply the distributive property.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{2} z^{2} y^{2} (4 x z^{2} y) - 8 x^{2} z^{2} y^{2} (- 8 x z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.2.1.2.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 (x \cdot x^{2}) z^{2} y^{2} (4 z^{2} y) - 8 x^{2} z^{2} y^{2} (- 8 x z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.2.2

Multiply $x$ by $x^{2}$ .

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Step 7.2.1.2.2.1

Raise $x$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 (x^{1} x^{2}) z^{2} y^{2} (4 z^{2} y) - 8 x^{2} z^{2} y^{2} (- 8 x z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{1 + 2} z^{2} y^{2} (4 z^{2} y) - 8 x^{2} z^{2} y^{2} (- 8 x z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.2.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{2} y^{2} (4 z^{2} y) - 8 x^{2} z^{2} y^{2} (- 8 x z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.2.1.3.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{2} y^{2} (4 z^{2} y) - 8 (x \cdot x^{2}) z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 7.2.1.3.2.1

Raise $x$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{2} y^{2} (4 z^{2} y) - 8 (x^{1} x^{2}) z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{2} y^{2} (4 z^{2} y) - 8 x^{1 + 2} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.3.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{2} y^{2} (4 z^{2} y) - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4

Simplify each term.

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Step 7.2.1.4.1

Multiply $z^{2}$ by $z^{2}$ by adding the exponents.

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Step 7.2.1.4.1.1

Move $z^{2}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} (z^{2} z^{2}) y^{2} (4 y) - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{2 + 2} y^{2} (4 y) - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.1.3

Add $2$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{4} y^{2} (4 y) - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.2

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 7.2.1.4.2.1

Move $y$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{4} (y \cdot y^{2}) \cdot 4 - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.2.2

Multiply $y$ by $y^{2}$ .

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Step 7.2.1.4.2.2.1

Raise $y$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{4} (y^{1} y^{2}) \cdot 4 - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{4} y^{1 + 2} \cdot 4 - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.2.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 8 x^{3} z^{4} y^{3} \cdot 4 - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.3

Multiply $4$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{2} y^{2} (- 8 z^{4} y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.4

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 7.2.1.4.4.1

Move $z^{4}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} (z^{4} z^{2}) y^{2} (- 8 y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{4 + 2} y^{2} (- 8 y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.4.3

Add $4$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{6} y^{2} (- 8 y) - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.5

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 7.2.1.4.5.1

Move $y$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{6} (y \cdot y^{2}) \cdot - 8 - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.5.2

Multiply $y$ by $y^{2}$ .

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Step 7.2.1.4.5.2.1

Raise $y$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{6} (y^{1} y^{2}) \cdot - 8 - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{6} y^{1 + 2} \cdot - 8 - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.5.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} - 8 x^{3} z^{6} y^{3} \cdot - 8 - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.4.6

Multiply $- 8$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{2} z^{3} y (- 8 x z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.2.1.5.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 (x \cdot x^{2}) z^{3} y (- 8 z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.5.2

Multiply $x$ by $x^{2}$ .

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Step 7.2.1.5.2.1

Raise $x$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 (x^{1} x^{2}) z^{3} y (- 8 z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{1 + 2} z^{3} y (- 8 z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.5.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{3} y (- 8 z^{3} y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.6

Multiply $z^{3}$ by $z^{3}$ by adding the exponents.

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Step 7.2.1.6.1

Move $z^{3}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} (z^{3} z^{3}) y (- 8 y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{3 + 3} y (- 8 y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.6.3

Add $3$ and $3$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{6} y (- 8 y^{2}))) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.7

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 7.2.1.7.1

Move $y^{2}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{6} (y^{2} y) \cdot - 8)) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.7.2

Multiply $y^{2}$ by $y$ .

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Step 7.2.1.7.2.1

Raise $y$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{6} (y^{2} y^{1}) \cdot - 8)) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{6} y^{2 + 1} \cdot - 8)) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.7.3

Add $2$ and $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (- 8 x^{3} z^{6} y^{3} \cdot - 8)) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.8

Multiply $- 8$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - (64 x^{3} z^{6} y^{3})) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.1.9

Multiply $64$ by $- 1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - 64 x^{3} z^{6} y^{3}) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.2

Combine the opposite terms in $- 32 x^{3} z^{4} y^{3} + 64 x^{3} z^{6} y^{3} - 64 x^{3} z^{6} y^{3}$ .

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Step 7.2.2.1

Subtract $64 x^{3} z^{6} y^{3}$ from $64 x^{3} z^{6} y^{3}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3} + 0) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 7.2.2.2

Add $- 32 x^{3} z^{4} y^{3}$ and $0$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y | \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$

Step 8

Evaluate $| \begin{array}{c} - 8 x^{2} z^{2} y^{2} & 4 x z^{2} y - 8 x z^{2} y^{3} \\ - 8 x^{2} z^{3} y & - 8 x z^{3} y^{2} \end{array} |$ .

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Step 8.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{2} z^{2} y^{2} (- 8 x z^{3} y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2

Simplify the determinant.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.2.1.1.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 (x \cdot x^{2}) z^{2} y^{2} (- 8 z^{3} y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1.1.2.1

Raise $x$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 (x^{1} x^{2}) z^{2} y^{2} (- 8 z^{3} y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{1 + 2} z^{2} y^{2} (- 8 z^{3} y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.1.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} z^{2} y^{2} (- 8 z^{3} y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.2

Multiply $z^{2}$ by $z^{3}$ by adding the exponents.

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Step 8.2.1.2.1

Move $z^{3}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} (z^{3} z^{2}) y^{2} (- 8 y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} z^{3 + 2} y^{2} (- 8 y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.2.3

Add $3$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} z^{5} y^{2} (- 8 y^{2}) - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.3

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 8.2.1.3.1

Move $y^{2}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} z^{5} (y^{2} y^{2}) \cdot - 8 - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} z^{5} y^{2 + 2} \cdot - 8 - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.3.3

Add $2$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (- 8 x^{3} z^{5} y^{4} \cdot - 8 - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.4

Multiply $- 8$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 x^{2} z^{3} y (4 x z^{2} y - 8 x z^{2} y^{3})))$

Step 8.2.1.5

Apply the distributive property.

Step 8.2.1.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.2.1.6.1

Move $x$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 (x \cdot x^{2}) z^{3} y (4 z^{2} y) - 8 x^{2} z^{3} y (- 8 x z^{2} y^{3})))$

Step 8.2.1.6.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1.6.2.1

Raise $x$ to the power of $1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 (x^{1} x^{2}) z^{3} y (4 z^{2} y) - 8 x^{2} z^{3} y (- 8 x z^{2} y^{3})))$

Step 8.2.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 x^{1 + 2} z^{3} y (4 z^{2} y) - 8 x^{2} z^{3} y (- 8 x z^{2} y^{3})))$

Step 8.2.1.6.3

Add $1$ and $2$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 x^{3} z^{3} y (4 z^{2} y) - 8 x^{2} z^{3} y (- 8 x z^{2} y^{3})))$

Step 8.2.1.7

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.2.1.7.1

Move $x$ .

Step 8.2.1.7.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1.7.2.1

Raise $x$ to the power of $1$ .

Step 8.2.1.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 8.2.1.7.3

Add $1$ and $2$ .

Step 8.2.1.8

Simplify each term.

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Step 8.2.1.8.1

Multiply $z^{3}$ by $z^{2}$ by adding the exponents.

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Step 8.2.1.8.1.1

Move $z^{2}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 x^{3} (z^{2} z^{3}) y (4 y) - 8 x^{3} z^{3} y (- 8 z^{2} y^{3})))$

Step 8.2.1.8.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 x^{3} z^{2 + 3} y (4 y) - 8 x^{3} z^{3} y (- 8 z^{2} y^{3})))$

Step 8.2.1.8.1.3

Add $2$ and $3$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 8 x^{3} z^{5} y (4 y) - 8 x^{3} z^{3} y (- 8 z^{2} y^{3})))$

Step 8.2.1.8.2

Multiply $y$ by $y$ by adding the exponents.

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Step 8.2.1.8.2.1

Move $y$ .

Step 8.2.1.8.2.2

Multiply $y$ by $y$ .

Step 8.2.1.8.3

Multiply $4$ by $- 8$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} - (- 32 x^{3} z^{5} y^{2} - 8 x^{3} z^{3} y (- 8 z^{2} y^{3})))$

Step 8.2.1.8.4

Multiply $z^{3}$ by $z^{2}$ by adding the exponents.

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Step 8.2.1.8.4.1

Move $z^{2}$ .

Step 8.2.1.8.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 8.2.1.8.4.3

Add $2$ and $3$ .

Step 8.2.1.8.5

Multiply $y$ by $y^{3}$ by adding the exponents.

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Step 8.2.1.8.5.1

Move $y^{3}$ .

Step 8.2.1.8.5.2

Multiply $y^{3}$ by $y$ .

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Step 8.2.1.8.5.2.1

Raise $y$ to the power of $1$ .

Step 8.2.1.8.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 8.2.1.8.5.3

Add $3$ and $1$ .

Step 8.2.1.8.6

Multiply $- 8$ by $- 8$ .

Step 8.2.1.9

Apply the distributive property.

Step 8.2.1.10

Multiply $- 32$ by $- 1$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} + 32 (x^{3} z^{5} y^{2}) - (64 x^{3} z^{5} y^{4}))$

Step 8.2.1.11

Multiply $64$ by $- 1$ .

Step 8.2.1.12

Remove parentheses.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (64 x^{3} z^{5} y^{4} + 32 x^{3} z^{5} y^{2} - 64 x^{3} z^{5} y^{4})$

Step 8.2.2

Combine the opposite terms in $64 x^{3} z^{5} y^{4} + 32 x^{3} z^{5} y^{2} - 64 x^{3} z^{5} y^{4}$ .

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Step 8.2.2.1

Subtract $64 x^{3} z^{5} y^{4}$ from $64 x^{3} z^{5} y^{4}$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2} + 0)$

Step 8.2.2.2

Add $32 x^{3} z^{5} y^{2}$ and $0$ .

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2} - 32 x^{2} z^{6} y^{2} - 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9

Simplify the determinant.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the distributive property.

$- 8 x^{3} z^{2} y (16 x^{2} z^{4} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2

Simplify.

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Step 9.1.2.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 9.1.2.1.1

Move $x^{2}$ .

$- 8 (x^{2} x^{3}) z^{2} y (16 z^{4} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{2 + 3} z^{2} y (16 z^{4} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.1.3

Add $2$ and $3$ .

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 9.1.2.2.1

Move $x^{2}$ .

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 (x^{2} x^{3}) z^{2} y (- 32 z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 x^{2 + 3} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.2.3

Add $2$ and $3$ .

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{3} z^{2} y (- 32 x^{2} z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.3

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 9.1.2.3.1

Move $x^{2}$ .

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 (x^{2} x^{3}) z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{2 + 3} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.2.3.3

Add $2$ and $3$ .

$- 8 x^{5} z^{2} y (16 z^{4} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3

Simplify each term.

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Step 9.1.3.1

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 9.1.3.1.1

Move $z^{4}$ .

$- 8 x^{5} (z^{4} z^{2}) y (16 y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{5} z^{4 + 2} y (16 y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.1.3

Add $4$ and $2$ .

$- 8 x^{5} z^{6} y (16 y^{2}) - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 9.1.3.2.1

Move $y^{2}$ .

$- 8 x^{5} z^{6} (y^{2} y) \cdot 16 - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.2.2

Multiply $y^{2}$ by $y$ .

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Step 9.1.3.2.2.1

Raise $y$ to the power of $1$ .

$- 8 x^{5} z^{6} (y^{2} y^{1}) \cdot 16 - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{5} z^{6} y^{2 + 1} \cdot 16 - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.2.3

Add $2$ and $1$ .

$- 8 x^{5} z^{6} y^{3} \cdot 16 - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.3

Multiply $16$ by $- 8$ .

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{2} y (- 32 z^{6} y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.4

Multiply $z^{2}$ by $z^{6}$ by adding the exponents.

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Step 9.1.3.4.1

Move $z^{6}$ .

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} (z^{6} z^{2}) y (- 32 y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{6 + 2} y (- 32 y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.4.3

Add $6$ and $2$ .

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{8} y (- 32 y^{2}) - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.5

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 9.1.3.5.1

Move $y^{2}$ .

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{8} (y^{2} y) \cdot - 32 - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.5.2

Multiply $y^{2}$ by $y$ .

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Step 9.1.3.5.2.1

Raise $y$ to the power of $1$ .

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{8} (y^{2} y^{1}) \cdot - 32 - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{8} y^{2 + 1} \cdot - 32 - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.5.3

Add $2$ and $1$ .

$- 128 x^{5} z^{6} y^{3} - 8 x^{5} z^{8} y^{3} \cdot - 32 - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.6

Multiply $- 32$ by $- 8$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{2} y (- 32 z^{4} y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.7

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 9.1.3.7.1

Move $z^{4}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} (z^{4} z^{2}) y (- 32 y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{4 + 2} y (- 32 y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.7.3

Add $4$ and $2$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{6} y (- 32 y^{4}) + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.8

Multiply $y$ by $y^{4}$ by adding the exponents.

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Step 9.1.3.8.1

Move $y^{4}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{6} (y^{4} y) \cdot - 32 + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.8.2

Multiply $y^{4}$ by $y$ .

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Step 9.1.3.8.2.1

Raise $y$ to the power of $1$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{6} (y^{4} y^{1}) \cdot - 32 + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{6} y^{4 + 1} \cdot - 32 + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.8.3

Add $4$ and $1$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 8 x^{5} z^{6} y^{5} \cdot - 32 + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.3.9

Multiply $- 32$ by $- 8$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{2} z^{2} y^{2} (- 32 x^{3} z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.4

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 9.1.4.1

Move $x^{3}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 (x^{3} x^{2}) z^{2} y^{2} (- 32 z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{3 + 2} z^{2} y^{2} (- 32 z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.4.3

Add $3$ and $2$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} z^{2} y^{2} (- 32 z^{4} y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.5

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 9.1.5.1

Move $z^{4}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} (z^{4} z^{2}) y^{2} (- 32 y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} z^{4 + 2} y^{2} (- 32 y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.5.3

Add $4$ and $2$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} z^{6} y^{2} (- 32 y^{3}) - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.6

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 9.1.6.1

Move $y^{3}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} z^{6} (y^{3} y^{2}) \cdot - 32 - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} z^{6} y^{3 + 2} \cdot - 32 - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.6.3

Add $3$ and $2$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} + 8 x^{5} z^{6} y^{5} \cdot - 32 - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.7

Multiply $- 32$ by $8$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{2} z^{3} y (32 x^{3} z^{5} y^{2})$

Step 9.1.8

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 9.1.8.1

Move $x^{3}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 (x^{3} x^{2}) z^{3} y (32 z^{5} y^{2})$

Step 9.1.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{3 + 2} z^{3} y (32 z^{5} y^{2})$

Step 9.1.8.3

Add $3$ and $2$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{3} y (32 z^{5} y^{2})$

Step 9.1.9

Multiply $z^{3}$ by $z^{5}$ by adding the exponents.

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Step 9.1.9.1

Move $z^{5}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} (z^{5} z^{3}) y (32 y^{2})$

Step 9.1.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{5 + 3} y (32 y^{2})$

Step 9.1.9.3

Add $5$ and $3$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{8} y (32 y^{2})$

Step 9.1.10

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 9.1.10.1

Move $y^{2}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{8} (y^{2} y) \cdot 32$

Step 9.1.10.2

Multiply $y^{2}$ by $y$ .

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Step 9.1.10.2.1

Raise $y$ to the power of $1$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{8} (y^{2} y^{1}) \cdot 32$

Step 9.1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{8} y^{2 + 1} \cdot 32$

Step 9.1.10.3

Add $2$ and $1$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 8 x^{5} z^{8} y^{3} \cdot 32$

Step 9.1.11

Multiply $32$ by $- 8$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{8} y^{3}$

Step 9.2

Combine the opposite terms in $- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{6} y^{5} - 256 x^{5} z^{8} y^{3}$ .

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Step 9.2.1

Subtract $256 x^{5} z^{6} y^{5}$ from $256 x^{5} z^{6} y^{5}$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} + 0 - 256 x^{5} z^{8} y^{3}$

Step 9.2.2

Add $- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3}$ and $0$ .

$- 128 x^{5} z^{6} y^{3} + 256 x^{5} z^{8} y^{3} - 256 x^{5} z^{8} y^{3}$

Step 9.2.3

Subtract $256 x^{5} z^{8} y^{3}$ from $256 x^{5} z^{8} y^{3}$ .

$- 128 x^{5} z^{6} y^{3} + 0$

Step 9.2.4

Add $- 128 x^{5} z^{6} y^{3}$ and $0$ .

$- 128 x^{5} z^{6} y^{3}$