Precalculus Examples

${(\frac{2 x^{- 3} y^{2}}{x y^{3} z^{0}})}^{- 2}$

Step 1

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

${(\frac{2 y^{2}}{x y^{3} z^{0} x^{3}})}^{- 2}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.1.1

Move $x^{3}$ .

${(\frac{2 y^{2}}{x^{3} x y^{3} z^{0}})}^{- 2}$

Step 2.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.1.2.1

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

${(\frac{2 y^{2}}{x^{3} x^{1} y^{3} z^{0}})}^{- 2}$

Step 2.1.2.2

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

${(\frac{2 y^{2}}{x^{3 + 1} y^{3} z^{0}})}^{- 2}$

Step 2.1.3

Add $3$ and $1$ .

${(\frac{2 y^{2}}{x^{4} y^{3} z^{0}})}^{- 2}$

Step 2.2

Simplify $x^{4} y^{3} z^{0}$ .

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

Step 3

Cancel the common factor of $y^{2}$ and $y^{3}$ .

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Step 3.1

Factor $y^{2}$ out of $2 y^{2}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $y^{2}$ out of $x^{4} y^{3}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the product rule to $\frac{x^{4} y}{2}$ .

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

Step 5.2

Apply the product rule to $x^{4} y$ .

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

Step 6

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 6.1

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

$\frac{x^{4 \cdot 2} y^{2}}{2^{2}}$

Step 6.2

Multiply $4$ by $2$ .

$\frac{x^{8} y^{2}}{2^{2}}$

Step 7

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

$\frac{x^{8} y^{2}}{4}$