Precalculus Examples

${(\frac{9 x^{3} y}{y^{- 3}})}^{\frac{1}{2}}$

Step 1

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

${(9 x^{3} y \cdot y^{3})}^{\frac{1}{2}}$

Step 2

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

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Step 2.1

Move $y^{3}$ .

${(9 x^{3} (y^{3} y))}^{\frac{1}{2}}$

Step 2.2

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

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Step 2.2.1

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

${(9 x^{3} (y^{3} y^{1}))}^{\frac{1}{2}}$

Step 2.2.2

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

${(9 x^{3} y^{3 + 1})}^{\frac{1}{2}}$

Step 2.3

Add $3$ and $1$ .

${(9 x^{3} y^{4})}^{\frac{1}{2}}$

Step 3

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

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Step 3.1

Apply the product rule to $9 x^{3} y^{4}$ .

${(9 x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 3.2

Apply the product rule to $9 x^{3}$ .

$9^{\frac{1}{2}} {(x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 4

Simplify the expression.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

${(3^{2})}^{\frac{1}{2}} {(x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 4.2

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

$3^{2 (\frac{1}{2})} {(x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$3^{2 (\frac{1}{2})} {(x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 5.2

Rewrite the expression.

$3^{1} {(x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 6

Evaluate the exponent.

$3 {(x^{3})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 7

Multiply the exponents in ${(x^{3})}^{\frac{1}{2}}$ .

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Step 7.1

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

$3 x^{3 (\frac{1}{2})} {(y^{4})}^{\frac{1}{2}}$

Step 7.2

Combine $3$ and $\frac{1}{2}$ .

$3 x^{\frac{3}{2}} {(y^{4})}^{\frac{1}{2}}$

Step 8

Multiply the exponents in ${(y^{4})}^{\frac{1}{2}}$ .

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Step 8.1

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

$3 x^{\frac{3}{2}} y^{4 (\frac{1}{2})}$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $4$ .

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

Step 8.2.2

Cancel the common factor.

$3 x^{\frac{3}{2}} y^{2 \cdot 2 \frac{1}{2}}$

Step 8.2.3

Rewrite the expression.

$3 x^{\frac{3}{2}} y^{2}$

Step 9

Reorder factors in $3 x^{\frac{3}{2}} y^{2}$ .

$3 y^{2} x^{\frac{3}{2}}$