Precalculus Examples

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

Step 1

Move $y^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Move $z^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

Move $y^{2}$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.3

Add $2$ and $1$ .

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

Step 3.2

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

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Step 3.2.1

Move $z^{3}$ .

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

Step 3.2.2

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

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

Step 3.2.3

Add $3$ and $2$ .

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

Step 4

Move $x^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Add $1$ and $2$ .

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

Step 6

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

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

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 $\frac{2 y^{3} z^{5}}{x^{3}}$ .

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

Step 7.2

Apply the product rule to $2 y^{3} z^{5}$ .

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

Step 7.3

Apply the product rule to $2 y^{3}$ .

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

Step 8

Simplify the numerator.

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Step 8.1

Raise $2$ to the power of $3$ .

$\frac{8 {(y^{3})}^{3} {(z^{5})}^{3}}{{(x^{3})}^{3}}$

Step 8.2

Multiply the exponents in ${(y^{3})}^{3}$ .

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Step 8.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 y^{3 \cdot 3} {(z^{5})}^{3}}{{(x^{3})}^{3}}$

Step 8.2.2

Multiply $3$ by $3$ .

$\frac{8 y^{9} {(z^{5})}^{3}}{{(x^{3})}^{3}}$

Step 8.3

Multiply the exponents in ${(z^{5})}^{3}$ .

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Step 8.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 y^{9} z^{5 \cdot 3}}{{(x^{3})}^{3}}$

Step 8.3.2

Multiply $5$ by $3$ .

$\frac{8 y^{9} z^{15}}{{(x^{3})}^{3}}$

Step 9

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 y^{9} z^{15}}{x^{3 \cdot 3}}$

Step 9.2

Multiply $3$ by $3$ .

$\frac{8 y^{9} z^{15}}{x^{9}}$