Simplify ((125x^4y^(3/2))^(1/3))÷((xy)^(1/2))

${(125 x^{4} y^{\frac{3}{2}})}^{\frac{1}{3}} \div {(x y)}^{\frac{1}{2}}$

Rewrite the division as a fraction.

$\frac{{(125 x^{4} y^{\frac{3}{2}})}^{\frac{1}{3}}}{{(x y)}^{\frac{1}{2}}}$

Simplify the numerator.

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

Apply the product rule to $125 x^{4} y^{\frac{3}{2}}$ .

$\frac{{(125 x^{4})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}}}{{(x y)}^{\frac{1}{2}}}$

Apply the product rule to $125 x^{4}$ .

$\frac{125^{\frac{1}{3}} {(x^{4})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}}}{{(x y)}^{\frac{1}{2}}}$

Rewrite $125$ as $5^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Combine $4$ and $\frac{1}{3}$ .

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Apply the product rule to $x y$ .

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

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

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

Multiply $x^{\frac{4}{3}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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

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

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

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

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $2$ .

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

Add $- 3$ and $8$ .

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

Cancel the common factor.

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

Divide $5 x^{\frac{5}{6}}$ by $1$ .

$5 x^{\frac{5}{6}}$

Simplify ((125x^4y^(3/2))^(1/3))÷((xy)^(1/2))

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