Precalculus Examples

${(\frac{3 p^{3} q^{- 1}}{9 p^{2} q^{- 2}})}^{2}$

Step 1

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

${(\frac{3 p^{3} q^{- 1} q^{2}}{9 p^{2}})}^{2}$

Step 2

Simplify the numerator.

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Step 2.1

Multiply $q^{- 1}$ by $q^{2}$ by adding the exponents.

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Step 2.1.1

Move $q^{2}$ .

${(\frac{3 p^{3} (q^{2} q^{- 1})}{9 p^{2}})}^{2}$

Step 2.1.2

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

${(\frac{3 p^{3} q^{2 - 1}}{9 p^{2}})}^{2}$

Step 2.1.3

Subtract $1$ from $2$ .

${(\frac{3 p^{3} q^{1}}{9 p^{2}})}^{2}$

Step 2.2

Simplify $3 p^{3} q^{1}$ .

${(\frac{3 p^{3} q}{9 p^{2}})}^{2}$

Step 3

Cancel the common factor of $3$ and $9$ .

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Step 3.1

Factor $3$ out of $3 p^{3} q$ .

${(\frac{3 (p^{3} q)}{9 p^{2}})}^{2}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $9 p^{2}$ .

${(\frac{3 (p^{3} q)}{3 (3 p^{2})})}^{2}$

Step 3.2.2

Cancel the common factor.

${(\frac{3 (p^{3} q)}{3 (3 p^{2})})}^{2}$

Step 3.2.3

Rewrite the expression.

${(\frac{p^{3} q}{3 p^{2}})}^{2}$

Step 4

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

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Step 4.1

Factor $p^{2}$ out of $p^{3} q$ .

${(\frac{p^{2} (p q)}{3 p^{2}})}^{2}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

${(\frac{p^{2} (p q)}{p^{2} \cdot 3})}^{2}$

Step 4.2.2

Cancel the common factor.

${(\frac{p^{2} (p q)}{p^{2} \cdot 3})}^{2}$

Step 4.2.3

Rewrite the expression.

${(\frac{p q}{3})}^{2}$

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 $\frac{p q}{3}$ .

$\frac{{(p q)}^{2}}{3^{2}}$

Step 5.2

Apply the product rule to $p q$ .

$\frac{p^{2} q^{2}}{3^{2}}$

Step 6

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

$\frac{p^{2} q^{2}}{9}$