Simplify ((x^(-1/2)y^2)^(-5/4))/(x^2y^(1/2))

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

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

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

Simplify the denominator.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Multiply $2$ by $4$ .

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

Combine exponents.

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $5$ and $1$ .

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

Divide $6$ by $2$ .

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

Reduce the expression by cancelling the common factors.

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $8$ .

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

Add $- 5$ and $16$ .

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

Simplify ((x^(-1/2)y^2)^(-5/4))/(x^2y^(1/2))

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