Precalculus Examples

$(\frac{x^{- \frac{5}{3}}}{y^{\frac{1}{5}}}) {(\frac{x^{- 5}}{y^{- 3}})}^{\frac{1}{15}}$

Step 1

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

$\frac{\frac{1}{x^{\frac{5}{3}}}}{y^{\frac{1}{5}}} {(\frac{x^{- 5}}{y^{- 3}})}^{\frac{1}{15}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{x^{\frac{5}{3}}} \cdot \frac{1}{y^{\frac{1}{5}}} {(\frac{x^{- 5}}{y^{- 3}})}^{\frac{1}{15}}$

Step 3

Combine fractions.

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Step 3.1

Combine.

$\frac{1 \cdot 1}{x^{\frac{5}{3}} y^{\frac{1}{5}}} {(\frac{x^{- 5}}{y^{- 3}})}^{\frac{1}{15}}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $1$ by $1$ .

$\frac{1}{x^{\frac{5}{3}} y^{\frac{1}{5}}} {(\frac{x^{- 5}}{y^{- 3}})}^{\frac{1}{15}}$

Step 3.2.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{\frac{5}{3}} y^{\frac{1}{5}}} {(\frac{1}{y^{- 3} x^{5}})}^{\frac{1}{15}}$

Step 3.2.3

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

$\frac{1}{x^{\frac{5}{3}} y^{\frac{1}{5}}} {(\frac{y^{3}}{x^{5}})}^{\frac{1}{15}}$

Step 3.2.4

Apply the product rule to $\frac{y^{3}}{x^{5}}$ .

$\frac{1}{x^{\frac{5}{3}} y^{\frac{1}{5}}} \cdot \frac{{(y^{3})}^{\frac{1}{15}}}{{(x^{5})}^{\frac{1}{15}}}$

Step 3.3

Combine.

$\frac{1 {(y^{3})}^{\frac{1}{15}}}{x^{\frac{5}{3}} y^{\frac{1}{5}} {(x^{5})}^{\frac{1}{15}}}$

Step 3.4

Multiply ${(y^{3})}^{\frac{1}{15}}$ by $1$ .

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{5}{3}} y^{\frac{1}{5}} {(x^{5})}^{\frac{1}{15}}}$

Step 4

Simplify the denominator.

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Step 4.1

Multiply the exponents in ${(x^{5})}^{\frac{1}{15}}$ .

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Step 4.1.1

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

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{5}{3}} y^{\frac{1}{5}} x^{5 (\frac{1}{15})}}$

Step 4.1.2

Cancel the common factor of $5$ .

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Step 4.1.2.1

Factor $5$ out of $15$ .

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{5}{3}} y^{\frac{1}{5}} x^{5 \frac{1}{5 (3)}}}$

Step 4.1.2.2

Cancel the common factor.

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{5}{3}} y^{\frac{1}{5}} x^{5 \frac{1}{5 \cdot 3}}}$

Step 4.1.2.3

Rewrite the expression.

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{5}{3}} y^{\frac{1}{5}} x^{\frac{1}{3}}}$

Step 4.2

Multiply $x^{\frac{5}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 4.2.1

Move $x^{\frac{1}{3}}$ .

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{1}{3}} x^{\frac{5}{3}} y^{\frac{1}{5}}}$

Step 4.2.2

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

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{1}{3} + \frac{5}{3}} y^{\frac{1}{5}}}$

Step 4.2.3

Combine the numerators over the common denominator.

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{1 + 5}{3}} y^{\frac{1}{5}}}$

Step 4.2.4

Add $1$ and $5$ .

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{\frac{6}{3}} y^{\frac{1}{5}}}$

Step 4.2.5

Divide $6$ by $3$ .

$\frac{{(y^{3})}^{\frac{1}{15}}}{x^{2} y^{\frac{1}{5}}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Multiply the exponents in ${(y^{3})}^{\frac{1}{15}}$ .

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Step 5.1.1

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

$\frac{y^{3 (\frac{1}{15})}}{x^{2} y^{\frac{1}{5}}}$

Step 5.1.2

Cancel the common factor of $3$ .

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Step 5.1.2.1

Factor $3$ out of $15$ .

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

Step 5.1.2.2

Cancel the common factor.

$\frac{y^{3 \frac{1}{3 \cdot 5}}}{x^{2} y^{\frac{1}{5}}}$

Step 5.1.2.3

Rewrite the expression.

$\frac{y^{\frac{1}{5}}}{x^{2} y^{\frac{1}{5}}}$

Step 5.2

Cancel the common factor.

$\frac{y^{\frac{1}{5}}}{x^{2} y^{\frac{1}{5}}}$

Step 5.3

Rewrite the expression.

$\frac{1}{x^{2}}$