Precalculus Examples

${(\frac{x y^{- 2} z^{- 3}}{x^{2} y^{3} z^{- 4}})}^{- 3}$

Step 1

Move

$y^{- 2}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{x z^{- 3}}{x^{2} y^{3} z^{- 4} y^{2}})}^{- 3}$

Step 2

Multiply

$y^{3}$ by

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

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Step 2.1

Move

$y^{2}$ .

${(\frac{x z^{- 3}}{x^{2} (y^{2} y^{3}) z^{- 4}})}^{- 3}$

Step 2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{x z^{- 3}}{x^{2} y^{2 + 3} z^{- 4}})}^{- 3}$

Step 2.3

Add

$2$ and

$3$ .

${(\frac{x z^{- 3}}{x^{2} y^{5} z^{- 4}})}^{- 3}$

Step 3

Move

$z^{- 4}$ to the numerator using the negative exponent rule

$\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{x z^{- 3} z^{4}}{x^{2} y^{5}})}^{- 3}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply

$z^{- 3}$ by

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

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Step 4.1.1

Move

$z^{4}$ .

${(\frac{x (z^{4} z^{- 3})}{x^{2} y^{5}})}^{- 3}$

Step 4.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{x z^{4 - 3}}{x^{2} y^{5}})}^{- 3}$

Step 4.1.3

Subtract

$3$ from

$4$ .

${(\frac{x z^{1}}{x^{2} y^{5}})}^{- 3}$

Step 4.2

Simplify

$x z^{1}$ .

${(\frac{x z}{x^{2} y^{5}})}^{- 3}$

Step 5

Cancel the common factor of

$x$ and

$x^{2}$ .

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Step 5.1

Factor

$x$ out of

$x z$ .

${(\frac{x (z)}{x^{2} y^{5}})}^{- 3}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor

$x$ out of

$x^{2} y^{5}$ .

${(\frac{x (z)}{x (x y^{5})})}^{- 3}$

Step 5.2.2

Cancel the common factor.

${(\frac{x z}{x (x y^{5})})}^{- 3}$

Step 5.2.3

Rewrite the expression.

${(\frac{z}{x y^{5}})}^{- 3}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{x y^{5}}{z})}^{3}$

Step 7

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1

Apply the product rule to

$\frac{x y^{5}}{z}$ .

$\frac{{(x y^{5})}^{3}}{z^{3}}$

Step 7.2

Apply the product rule to

$x y^{5}$ .

$\frac{x^{3} {(y^{5})}^{3}}{z^{3}}$

Step 8

Multiply the exponents in

${(y^{5})}^{3}$ .

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Step 8.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{3} y^{5 \cdot 3}}{z^{3}}$

Step 8.2

Multiply

$5$ by

$3$ .

$\frac{x^{3} y^{15}}{z^{3}}$