Algebra Examples

${(4 z)}^{- 3} \cdot {(2 z)}^{5}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(4 z)}^{3}} \cdot {(2 z)}^{5}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $4 z$ .

$\frac{1}{4^{3} z^{3}} \cdot {(2 z)}^{5}$

Step 2.2

Raise $4$ to the power of $3$ .

$\frac{1}{64 z^{3}} \cdot {(2 z)}^{5}$

Step 3

Simplify terms.

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Step 3.1

Apply the product rule to $2 z$ .

$\frac{1}{64 z^{3}} \cdot (2^{5} z^{5})$

Step 3.2

Simplify the expression.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$2^{5} \frac{1}{64 z^{3}} z^{5}$

Step 3.2.2

Raise $2$ to the power of $5$ .

$32 \frac{1}{64 z^{3}} z^{5}$

Step 3.3

Cancel the common factor of $32$ .

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Step 3.3.1

Factor $32$ out of $64 z^{3}$ .

$32 \frac{1}{32 (2 z^{3})} z^{5}$

Step 3.3.2

Cancel the common factor.

$32 \frac{1}{32 (2 z^{3})} z^{5}$

Step 3.3.3

Rewrite the expression.

$\frac{1}{2 z^{3}} z^{5}$

Step 3.4

Cancel the common factor of $z^{3}$ .

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Step 3.4.1

Factor $z^{3}$ out of $2 z^{3}$ .

$\frac{1}{z^{3} \cdot 2} z^{5}$

Step 3.4.2

Factor $z^{3}$ out of $z^{5}$ .

$\frac{1}{z^{3} \cdot 2} (z^{3} z^{2})$

Step 3.4.3

Cancel the common factor.

$\frac{1}{z^{3} \cdot 2} (z^{3} z^{2})$

Step 3.4.4

Rewrite the expression.

$\frac{1}{2} z^{2}$

Step 3.5

Combine $\frac{1}{2}$ and $z^{2}$ .

$\frac{z^{2}}{2}$