Basic Math Examples

$(11 y^{2} - 9 y z) (8 z^{2} - 11 y z + 7 y^{2})$

Step 1

Expand $(11 y^{2} - 9 y z) (8 z^{2} - 11 y z + 7 y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$11 y^{2} (8 z^{2}) + 11 y^{2} (- 11 y z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$11 \cdot 8 y^{2} z^{2} + 11 y^{2} (- 11 y z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.2

Multiply $11$ by $8$ .

$88 y^{2} z^{2} + 11 y^{2} (- 11 y z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.3

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

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Step 2.1.3.1

Move $y$ .

$88 y^{2} z^{2} + 11 (y \cdot y^{2}) (- 11 z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.3.2

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

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Step 2.1.3.2.1

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

$88 y^{2} z^{2} + 11 (y^{1} y^{2}) (- 11 z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.3.2.2

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

$88 y^{2} z^{2} + 11 y^{1 + 2} (- 11 z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.3.3

Add $1$ and $2$ .

$88 y^{2} z^{2} + 11 y^{3} (- 11 z) + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$88 y^{2} z^{2} + 11 \cdot - 11 y^{3} z + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.5

Multiply $11$ by $- 11$ .

$88 y^{2} z^{2} - 121 y^{3} z + 11 y^{2} (7 y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$88 y^{2} z^{2} - 121 y^{3} z + 11 \cdot 7 y^{2} y^{2} - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.7

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

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Step 2.1.7.1

Move $y^{2}$ .

$88 y^{2} z^{2} - 121 y^{3} z + 11 \cdot 7 (y^{2} y^{2}) - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.7.2

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

$88 y^{2} z^{2} - 121 y^{3} z + 11 \cdot 7 y^{2 + 2} - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.7.3

Add $2$ and $2$ .

$88 y^{2} z^{2} - 121 y^{3} z + 11 \cdot 7 y^{4} - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.8

Multiply $11$ by $7$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 9 y z (8 z^{2}) - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.9

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

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Step 2.1.9.1

Move $z^{2}$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 9 y (z^{2} z) \cdot 8 - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 9 y (z^{2} z^{1}) \cdot 8 - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.9.2.2

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

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 9 y z^{2 + 1} \cdot 8 - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.9.3

Add $2$ and $1$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 9 y z^{3} \cdot 8 - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.10

Multiply $8$ by $- 9$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} - 9 y z (- 11 y z) - 9 y z (7 y^{2})$

Step 2.1.11

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

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Step 2.1.11.1

Move $y$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} - 9 (y \cdot y) z (- 11 z) - 9 y z (7 y^{2})$

Step 2.1.11.2

Multiply $y$ by $y$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} - 9 y^{2} z (- 11 z) - 9 y z (7 y^{2})$

Step 2.1.12

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

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Step 2.1.12.1

Move $z$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} - 9 y^{2} (z \cdot z) \cdot - 11 - 9 y z (7 y^{2})$

Step 2.1.12.2

Multiply $z$ by $z$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} - 9 y^{2} z^{2} \cdot - 11 - 9 y z (7 y^{2})$

Step 2.1.13

Multiply $- 11$ by $- 9$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 99 y^{2} z^{2} - 9 y z (7 y^{2})$

Step 2.1.14

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

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Step 2.1.14.1

Move $y^{2}$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 99 y^{2} z^{2} - 9 (y^{2} y) z \cdot 7$

Step 2.1.14.2

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

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Step 2.1.14.2.1

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

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 99 y^{2} z^{2} - 9 (y^{2} y^{1}) z \cdot 7$

Step 2.1.14.2.2

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

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 99 y^{2} z^{2} - 9 y^{2 + 1} z \cdot 7$

Step 2.1.14.3

Add $2$ and $1$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 99 y^{2} z^{2} - 9 y^{3} z \cdot 7$

Step 2.1.15

Multiply $7$ by $- 9$ .

$88 y^{2} z^{2} - 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 99 y^{2} z^{2} - 63 y^{3} z$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $88 y^{2} z^{2}$ and $99 y^{2} z^{2}$ .

$- 121 y^{3} z + 77 y^{4} - 72 y z^{3} + 187 y^{2} z^{2} - 63 y^{3} z$

Step 2.2.2

Subtract $63 y^{3} z$ from $- 121 y^{3} z$ .

$77 y^{4} - 72 y z^{3} + 187 y^{2} z^{2} - 184 y^{3} z$