Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $3 (\frac{3}{2})$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $3$ by $3$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2

Move the negative in front of the fraction.

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

Step 2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4

Combine.

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

Step 2.5

Multiply $1$ by $1$ .

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

Step 2.6

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

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Step 2.6.1

Apply the product rule to $\frac{1}{x y^{\frac{2}{3}}}$ .

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

Step 2.6.2

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

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

Step 2.7

One to any power is one.

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

Step 2.8

Multiply the exponents in ${(y^{\frac{2}{3}})}^{\frac{1}{4}}$ .

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Step 2.8.1

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

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

Step 2.8.2

Cancel the common factor of $2$ .

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Step 2.8.2.1

Factor $2$ out of $4$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.8.3

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

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

Step 2.8.4

Multiply $3$ by $2$ .

$\frac{x^{\frac{9}{2}} y^{3}}{\frac{1}{x^{\frac{1}{4}} y^{\frac{1}{6}}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$x^{\frac{9}{2}} y^{3} (x^{\frac{1}{4}} y^{\frac{1}{6}})$

Step 4

Multiply $x^{\frac{9}{2}}$ by $x^{\frac{1}{4}}$ by adding the exponents.

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Step 4.1

Move $x^{\frac{1}{4}}$ .

$x^{\frac{1}{4}} x^{\frac{9}{2}} y^{3} y^{\frac{1}{6}}$

Step 4.2

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

$x^{\frac{1}{4} + \frac{9}{2}} y^{3} y^{\frac{1}{6}}$

Step 4.3

To write $\frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{1}{4} + \frac{9}{2} \cdot \frac{2}{2}} y^{3} y^{\frac{1}{6}}$

Step 4.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$x^{\frac{1}{4} + \frac{9 \cdot 2}{2 \cdot 2}} y^{3} y^{\frac{1}{6}}$

Step 4.4.2

Multiply $2$ by $2$ .

$x^{\frac{1}{4} + \frac{9 \cdot 2}{4}} y^{3} y^{\frac{1}{6}}$

Step 4.5

Combine the numerators over the common denominator.

$x^{\frac{1 + 9 \cdot 2}{4}} y^{3} y^{\frac{1}{6}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $9$ by $2$ .

$x^{\frac{1 + 18}{4}} y^{3} y^{\frac{1}{6}}$

Step 4.6.2

Add $1$ and $18$ .

$x^{\frac{19}{4}} y^{3} y^{\frac{1}{6}}$

Step 5

Multiply $y^{3}$ by $y^{\frac{1}{6}}$ by adding the exponents.

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Step 5.1

Move $y^{\frac{1}{6}}$ .

$x^{\frac{19}{4}} (y^{\frac{1}{6}} y^{3})$

Step 5.2

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

$x^{\frac{19}{4}} y^{\frac{1}{6} + 3}$

Step 5.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x^{\frac{19}{4}} y^{\frac{1}{6} + 3 \cdot \frac{6}{6}}$

Step 5.4

Combine $3$ and $\frac{6}{6}$ .

$x^{\frac{19}{4}} y^{\frac{1}{6} + \frac{3 \cdot 6}{6}}$

Step 5.5

Combine the numerators over the common denominator.

$x^{\frac{19}{4}} y^{\frac{1 + 3 \cdot 6}{6}}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $3$ by $6$ .

$x^{\frac{19}{4}} y^{\frac{1 + 18}{6}}$

Step 5.6.2

Add $1$ and $18$ .

$x^{\frac{19}{4}} y^{\frac{19}{6}}$