Algebra Examples

$\frac{{(x^{- \frac{1}{2}} y^{2})}^{- \frac{5}{4}}}{x^{2} y^{\frac{1}{2}}}$

Step 1

Move ${(x^{- \frac{1}{2}} y^{2})}^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2} y^{\frac{1}{2}} {(x^{- \frac{1}{2}} y^{2})}^{\frac{5}{4}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{1}{x^{2} y^{\frac{1}{2}} {(\frac{1}{x^{\frac{1}{2}}} y^{2})}^{\frac{5}{4}}}$

Step 2.2

Combine $\frac{1}{x^{\frac{1}{2}}}$ and $y^{2}$ .

$\frac{1}{x^{2} y^{\frac{1}{2}} {(\frac{y^{2}}{x^{\frac{1}{2}}})}^{\frac{5}{4}}}$

Step 2.3

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

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{{(y^{2})}^{\frac{5}{4}}}{{(x^{\frac{1}{2}})}^{\frac{5}{4}}}}$

Step 2.4

Multiply the exponents in ${(y^{2})}^{\frac{5}{4}}$ .

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Step 2.4.1

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

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{2 (\frac{5}{4})}}{{(x^{\frac{1}{2}})}^{\frac{5}{4}}}}$

Step 2.4.2

Cancel the common factor of $2$ .

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Step 2.4.2.1

Factor $2$ out of $4$ .

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{2 \frac{5}{2 (2)}}}{{(x^{\frac{1}{2}})}^{\frac{5}{4}}}}$

Step 2.4.2.2

Cancel the common factor.

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{2 \frac{5}{2 \cdot 2}}}{{(x^{\frac{1}{2}})}^{\frac{5}{4}}}}$

Step 2.4.2.3

Rewrite the expression.

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{\frac{5}{2}}}{{(x^{\frac{1}{2}})}^{\frac{5}{4}}}}$

Step 2.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{\frac{5}{4}}$ .

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Step 2.5.1

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

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{\frac{5}{2}}}{x^{\frac{1}{2} \cdot \frac{5}{4}}}}$

Step 2.5.2

Multiply $\frac{1}{2} \cdot \frac{5}{4}$ .

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Step 2.5.2.1

Multiply $\frac{1}{2}$ by $\frac{5}{4}$ .

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{\frac{5}{2}}}{x^{\frac{5}{2 \cdot 4}}}}$

Step 2.5.2.2

Multiply $2$ by $4$ .

$\frac{1}{x^{2} y^{\frac{1}{2}} \frac{y^{\frac{5}{2}}}{x^{\frac{5}{8}}}}$

Step 2.6

Combine exponents.

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Step 2.6.1

Combine $x^{2}$ and $\frac{y^{\frac{5}{2}}}{x^{\frac{5}{8}}}$ .

$\frac{1}{y^{\frac{1}{2}} \frac{x^{2} y^{\frac{5}{2}}}{x^{\frac{5}{8}}}}$

Step 2.6.2

Combine $y^{\frac{1}{2}}$ and $\frac{x^{2} y^{\frac{5}{2}}}{x^{\frac{5}{8}}}$ .

$\frac{1}{\frac{y^{\frac{1}{2}} (x^{2} y^{\frac{5}{2}})}{x^{\frac{5}{8}}}}$

Step 2.6.3

Multiply $y^{\frac{1}{2}}$ by $y^{\frac{5}{2}}$ by adding the exponents.

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Step 2.6.3.1

Move $y^{\frac{5}{2}}$ .

$\frac{1}{\frac{y^{\frac{5}{2}} y^{\frac{1}{2}} x^{2}}{x^{\frac{5}{8}}}}$

Step 2.6.3.2

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

$\frac{1}{\frac{y^{\frac{5}{2} + \frac{1}{2}} x^{2}}{x^{\frac{5}{8}}}}$

Step 2.6.3.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{y^{\frac{5 + 1}{2}} x^{2}}{x^{\frac{5}{8}}}}$

Step 2.6.3.4

Add $5$ and $1$ .

$\frac{1}{\frac{y^{\frac{6}{2}} x^{2}}{x^{\frac{5}{8}}}}$

Step 2.6.3.5

Divide $6$ by $2$ .

$\frac{1}{\frac{y^{3} x^{2}}{x^{\frac{5}{8}}}}$

Step 2.7

Reduce the expression by cancelling the common factors.

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Step 2.7.1

Move $x^{\frac{5}{8}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{1}{y^{3} x^{2} x^{- \frac{5}{8}}}$

Step 2.7.2

Multiply $x^{2}$ by $x^{- \frac{5}{8}}$ by adding the exponents.

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Step 2.7.2.1

Move $x^{- \frac{5}{8}}$ .

$\frac{1}{y^{3} (x^{- \frac{5}{8}} x^{2})}$

Step 2.7.2.2

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

$\frac{1}{y^{3} x^{- \frac{5}{8} + 2}}$

Step 2.7.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{y^{3} x^{- \frac{5}{8} + 2 \cdot \frac{8}{8}}}$

Step 2.7.2.4

Combine $2$ and $\frac{8}{8}$ .

$\frac{1}{y^{3} x^{- \frac{5}{8} + \frac{2 \cdot 8}{8}}}$

Step 2.7.2.5

Combine the numerators over the common denominator.

$\frac{1}{y^{3} x^{\frac{- 5 + 2 \cdot 8}{8}}}$

Step 2.7.2.6

Simplify the numerator.

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Step 2.7.2.6.1

Multiply $2$ by $8$ .

$\frac{1}{y^{3} x^{\frac{- 5 + 16}{8}}}$

Step 2.7.2.6.2

Add $- 5$ and $16$ .

$\frac{1}{y^{3} x^{\frac{11}{8}}}$