Simplify ((x^(-1/2)y^4)^(1/4))/(x^(2/3)y^(3/2)*x^(-3/2)y^(1/2))

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

Step 1

Multiply $x^{\frac{2}{3}}$ by $x^{- \frac{3}{2}}$ by adding the exponents.

Move $x^{- \frac{3}{2}}$ .

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

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

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

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $3$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{3 \cdot 3}{6} + \frac{2}{3} \cdot \frac{2}{2}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

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

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{3 \cdot 3}{6} + \frac{2 \cdot 2}{3 \cdot 2}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Multiply $3$ by $2$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{3 \cdot 3}{6} + \frac{2 \cdot 2}{6}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{\frac{- 3 \cdot 3 + 2 \cdot 2}{6}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Simplify the numerator.

Multiply $- 3$ by $3$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{\frac{- 9 + 2 \cdot 2}{6}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Multiply $2$ by $2$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{\frac{- 9 + 4}{6}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Add $- 9$ and $4$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{\frac{- 5}{6}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Move the negative in front of the fraction.

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{5}{6}} y^{\frac{3}{2}} \cdot y^{\frac{1}{2}}}$

Step 2

Multiply $y^{\frac{3}{2}}$ by $y^{\frac{1}{2}}$ by adding the exponents.

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

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{5}{6}} (y^{\frac{1}{2}} y^{\frac{3}{2}})}$

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

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{5}{6}} y^{\frac{1}{2} + \frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{5}{6}} y^{\frac{1 + 3}{2}}}$

Add $1$ and $3$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{5}{6}} y^{\frac{4}{2}}}$

Divide $4$ by $2$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}}}{x^{- \frac{5}{6}} y^{2}}$

Step 3

Move $x^{- \frac{5}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(x^{- \frac{1}{2}} y^{4})}^{\frac{1}{4}} x^{\frac{5}{6}}}{y^{2}}$

Step 4

Simplify the numerator.

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

$\frac{{(\frac{1}{x^{\frac{1}{2}}} y^{4})}^{\frac{1}{4}} x^{\frac{5}{6}}}{y^{2}}$

Combine $\frac{1}{x^{\frac{1}{2}}}$ and $y^{4}$ .

$\frac{{(\frac{y^{4}}{x^{\frac{1}{2}}})}^{\frac{1}{4}} x^{\frac{5}{6}}}{y^{2}}$

Apply the product rule to $\frac{y^{4}}{x^{\frac{1}{2}}}$ .

$\frac{\frac{{(y^{4})}^{\frac{1}{4}}}{{(x^{\frac{1}{2}})}^{\frac{1}{4}}} x^{\frac{5}{6}}}{y^{2}}$

Simplify the numerator.

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

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

$\frac{\frac{y^{4 (\frac{1}{4})}}{{(x^{\frac{1}{2}})}^{\frac{1}{4}}} x^{\frac{5}{6}}}{y^{2}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\frac{y^{4 (\frac{1}{4})}}{{(x^{\frac{1}{2}})}^{\frac{1}{4}}} x^{\frac{5}{6}}}{y^{2}}$

Rewrite the expression.

$\frac{\frac{y^{1}}{{(x^{\frac{1}{2}})}^{\frac{1}{4}}} x^{\frac{5}{6}}}{y^{2}}$

Simplify.

$\frac{\frac{y}{{(x^{\frac{1}{2}})}^{\frac{1}{4}}} x^{\frac{5}{6}}}{y^{2}}$

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

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

$\frac{\frac{y}{x^{\frac{1}{2} \cdot \frac{1}{4}}} x^{\frac{5}{6}}}{y^{2}}$

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

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

$\frac{\frac{y}{x^{\frac{1}{2 \cdot 4}}} x^{\frac{5}{6}}}{y^{2}}$

Multiply $2$ by $4$ .

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

Step 5

Combine $\frac{y}{x^{\frac{1}{8}}}$ and $x^{\frac{5}{6}}$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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

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

Multiply $x^{\frac{5}{6}}$ by $x^{- \frac{1}{8}}$ by adding the exponents.

Move $x^{- \frac{1}{8}}$ .

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

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

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

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y x^{- \frac{1}{8} \cdot \frac{3}{3} + \frac{5}{6}}}{y^{2}}$

To write $\frac{5}{6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y x^{- \frac{1}{8} \cdot \frac{3}{3} + \frac{5}{6} \cdot \frac{4}{4}}}{y^{2}}$

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$\frac{y x^{- \frac{3}{8 \cdot 3} + \frac{5}{6} \cdot \frac{4}{4}}}{y^{2}}$

Multiply $8$ by $3$ .

$\frac{y x^{- \frac{3}{24} + \frac{5}{6} \cdot \frac{4}{4}}}{y^{2}}$

Multiply $\frac{5}{6}$ by $\frac{4}{4}$ .

$\frac{y x^{- \frac{3}{24} + \frac{5 \cdot 4}{6 \cdot 4}}}{y^{2}}$

Multiply $6$ by $4$ .

$\frac{y x^{- \frac{3}{24} + \frac{5 \cdot 4}{24}}}{y^{2}}$

Combine the numerators over the common denominator.

$\frac{y x^{\frac{- 3 + 5 \cdot 4}{24}}}{y^{2}}$

Simplify the numerator.

Multiply $5$ by $4$ .

$\frac{y x^{\frac{- 3 + 20}{24}}}{y^{2}}$

Add $- 3$ and $20$ .

$\frac{y x^{\frac{17}{24}}}{y^{2}}$

Step 7

Factor $y$ out of $y x^{\frac{17}{24}}$ .

$\frac{y (x^{\frac{17}{24}})}{y^{2}}$

Step 8

Cancel the common factors.

Factor $y$ out of $y^{2}$ .

$\frac{y (x^{\frac{17}{24}})}{y \cdot y}$

Cancel the common factor.

$\frac{y x^{\frac{17}{24}}}{y \cdot y}$

Rewrite the expression.

$\frac{x^{\frac{17}{24}}}{y}$

Simplify ((x^(-1/2)y^4)^(1/4))/(x^(2/3)y^(3/2)*x^(-3/2)y^(1/2))

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