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

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

Step 1

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

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

Step 2

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

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

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

Multiply $x^{- \frac{1}{4}}$ by $x$ .

Raise $x$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $4$ .

${(\frac{4 y^{2}}{x^{\frac{3}{4}} y^{\frac{1}{5}}})}^{\frac{1}{2}}$

Step 3

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

${(\frac{4 y^{2} y^{- \frac{1}{5}}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

Step 4

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

Move $y^{- \frac{1}{5}}$ .

${(\frac{4 (y^{- \frac{1}{5}} y^{2})}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

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

${(\frac{4 y^{- \frac{1}{5} + 2}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(\frac{4 y^{- \frac{1}{5} + 2 \cdot \frac{5}{5}}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

Combine $2$ and $\frac{5}{5}$ .

${(\frac{4 y^{- \frac{1}{5} + \frac{2 \cdot 5}{5}}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

Combine the numerators over the common denominator.

${(\frac{4 y^{\frac{- 1 + 2 \cdot 5}{5}}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

Simplify the numerator.

Multiply $2$ by $5$ .

${(\frac{4 y^{\frac{- 1 + 10}{5}}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

Add $- 1$ and $10$ .

${(\frac{4 y^{\frac{9}{5}}}{x^{\frac{3}{4}}})}^{\frac{1}{2}}$

Step 5

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

Apply the product rule to $\frac{4 y^{\frac{9}{5}}}{x^{\frac{3}{4}}}$ .

$\frac{{(4 y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Apply the product rule to $4 y^{\frac{9}{5}}$ .

$\frac{4^{\frac{1}{2}} {(y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Step 6

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{{(2^{2})}^{\frac{1}{2}} {(y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

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

$\frac{2^{2 (\frac{1}{2})} {(y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2^{2 (\frac{1}{2})} {(y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{2^{1} {(y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Evaluate the exponent.

$\frac{2 {(y^{\frac{9}{5}})}^{\frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Multiply the exponents in ${(y^{\frac{9}{5}})}^{\frac{1}{2}}$ .

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

$\frac{2 y^{\frac{9}{5} \cdot \frac{1}{2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Multiply $\frac{9}{5} \cdot \frac{1}{2}$ .

Multiply $\frac{9}{5}$ by $\frac{1}{2}$ .

$\frac{2 y^{\frac{9}{5 \cdot 2}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Multiply $5$ by $2$ .

$\frac{2 y^{\frac{9}{10}}}{{(x^{\frac{3}{4}})}^{\frac{1}{2}}}$

Step 7

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

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

$\frac{2 y^{\frac{9}{10}}}{x^{\frac{3}{4} \cdot \frac{1}{2}}}$

Multiply $\frac{3}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

$\frac{2 y^{\frac{9}{10}}}{x^{\frac{3}{4 \cdot 2}}}$

Multiply $4$ by $2$ .

$\frac{2 y^{\frac{9}{10}}}{x^{\frac{3}{8}}}$

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

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