Basic Math Examples

${({(2 z - 1)}^{2} - 1)}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(2 z - 1)}^{2}$ as $(2 z - 1) (2 z - 1)$ .

${((2 z - 1) (2 z - 1) - 1)}^{2}$

Step 1.1.2

Expand $(2 z - 1) (2 z - 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

${(2 z (2 z - 1) - 1 (2 z - 1) - 1)}^{2}$

Step 1.1.2.2

Apply the distributive property.

${(2 z (2 z) + 2 z \cdot - 1 - 1 (2 z - 1) - 1)}^{2}$

Step 1.1.2.3

Apply the distributive property.

${(2 z (2 z) + 2 z \cdot - 1 - 1 (2 z) - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Rewrite using the commutative property of multiplication.

${(2 \cdot 2 z \cdot z + 2 z \cdot - 1 - 1 (2 z) - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Move $z$ .

${(2 \cdot 2 (z \cdot z) + 2 z \cdot - 1 - 1 (2 z) - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3.1.2.2

Multiply $z$ by $z$ .

${(2 \cdot 2 z^{2} + 2 z \cdot - 1 - 1 (2 z) - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3.1.3

Multiply $2$ by $2$ .

${(4 z^{2} + 2 z \cdot - 1 - 1 (2 z) - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3.1.4

Multiply $- 1$ by $2$ .

${(4 z^{2} - 2 z - 1 (2 z) - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3.1.5

Multiply $2$ by $- 1$ .

${(4 z^{2} - 2 z - 2 z - 1 \cdot - 1 - 1)}^{2}$

Step 1.1.3.1.6

Multiply $- 1$ by $- 1$ .

${(4 z^{2} - 2 z - 2 z + 1 - 1)}^{2}$

Step 1.1.3.2

Subtract $2 z$ from $- 2 z$ .

${(4 z^{2} - 4 z + 1 - 1)}^{2}$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Combine the opposite terms in $4 z^{2} - 4 z + 1 - 1$ .

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Step 1.2.1.1

Subtract $1$ from $1$ .

${(4 z^{2} - 4 z + 0)}^{2}$

Step 1.2.1.2

Add $4 z^{2} - 4 z$ and $0$ .

${(4 z^{2} - 4 z)}^{2}$

Step 1.2.2

Rewrite ${(4 z^{2} - 4 z)}^{2}$ as $(4 z^{2} - 4 z) (4 z^{2} - 4 z)$ .

$(4 z^{2} - 4 z) (4 z^{2} - 4 z)$

Step 2

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

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Step 2.1

Apply the distributive property.

$4 z^{2} (4 z^{2} - 4 z) - 4 z (4 z^{2} - 4 z)$

Step 2.2

Apply the distributive property.

$4 z^{2} (4 z^{2}) + 4 z^{2} (- 4 z) - 4 z (4 z^{2} - 4 z)$

Step 2.3

Apply the distributive property.

$4 z^{2} (4 z^{2}) + 4 z^{2} (- 4 z) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$4 \cdot 4 z^{2} z^{2} + 4 z^{2} (- 4 z) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.2

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

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Step 3.1.2.1

Move $z^{2}$ .

$4 \cdot 4 (z^{2} z^{2}) + 4 z^{2} (- 4 z) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.2.2

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

$4 \cdot 4 z^{2 + 2} + 4 z^{2} (- 4 z) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.2.3

Add $2$ and $2$ .

$4 \cdot 4 z^{4} + 4 z^{2} (- 4 z) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.3

Multiply $4$ by $4$ .

$16 z^{4} + 4 z^{2} (- 4 z) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$16 z^{4} + 4 \cdot - 4 z^{2} z - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.5

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

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Step 3.1.5.1

Move $z$ .

$16 z^{4} + 4 \cdot - 4 (z \cdot z^{2}) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.5.2

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

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Step 3.1.5.2.1

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

$16 z^{4} + 4 \cdot - 4 (z^{1} z^{2}) - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.5.2.2

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

$16 z^{4} + 4 \cdot - 4 z^{1 + 2} - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.5.3

Add $1$ and $2$ .

$16 z^{4} + 4 \cdot - 4 z^{3} - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.6

Multiply $4$ by $- 4$ .

$16 z^{4} - 16 z^{3} - 4 z (4 z^{2}) - 4 z (- 4 z)$

Step 3.1.7

Rewrite using the commutative property of multiplication.

$16 z^{4} - 16 z^{3} - 4 \cdot 4 z \cdot z^{2} - 4 z (- 4 z)$

Step 3.1.8

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

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Step 3.1.8.1

Move $z^{2}$ .

$16 z^{4} - 16 z^{3} - 4 \cdot 4 (z^{2} z) - 4 z (- 4 z)$

Step 3.1.8.2

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

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Step 3.1.8.2.1

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

$16 z^{4} - 16 z^{3} - 4 \cdot 4 (z^{2} z^{1}) - 4 z (- 4 z)$

Step 3.1.8.2.2

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

$16 z^{4} - 16 z^{3} - 4 \cdot 4 z^{2 + 1} - 4 z (- 4 z)$

Step 3.1.8.3

Add $2$ and $1$ .

$16 z^{4} - 16 z^{3} - 4 \cdot 4 z^{3} - 4 z (- 4 z)$

Step 3.1.9

Multiply $- 4$ by $4$ .

$16 z^{4} - 16 z^{3} - 16 z^{3} - 4 z (- 4 z)$

Step 3.1.10

Rewrite using the commutative property of multiplication.

$16 z^{4} - 16 z^{3} - 16 z^{3} - 4 \cdot - 4 z \cdot z$

Step 3.1.11

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

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Step 3.1.11.1

Move $z$ .

$16 z^{4} - 16 z^{3} - 16 z^{3} - 4 \cdot - 4 (z \cdot z)$

Step 3.1.11.2

Multiply $z$ by $z$ .

$16 z^{4} - 16 z^{3} - 16 z^{3} - 4 \cdot - 4 z^{2}$

Step 3.1.12

Multiply $- 4$ by $- 4$ .

$16 z^{4} - 16 z^{3} - 16 z^{3} + 16 z^{2}$

Step 3.2

Subtract $16 z^{3}$ from $- 16 z^{3}$ .

$16 z^{4} - 32 z^{3} + 16 z^{2}$