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

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

Move $x^{- \frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Simplify the numerator.

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

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

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine exponents.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $x^{1} y^{\frac{1}{4}}$ .

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

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

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

Multiply $y^{\frac{1}{2}}$ by $y^{- \frac{1}{4}}$ by adding the exponents.

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

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

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

$\frac{x}{y^{\frac{1}{4}}}$

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

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