Simplify ((32x^4y^4)/(4x^-2y))^(2/3)

${(\frac{32 x^{4} y^{4}}{4 x^{- 2} y})}^{\frac{2}{3}}$

Step 1

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

${(\frac{32 x^{4} y^{4} x^{2}}{4 y})}^{\frac{2}{3}}$

Step 2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

${(\frac{32 (x^{2} x^{4}) y^{4}}{4 y})}^{\frac{2}{3}}$

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

${(\frac{32 x^{2 + 4} y^{4}}{4 y})}^{\frac{2}{3}}$

Add $2$ and $4$ .

${(\frac{32 x^{6} y^{4}}{4 y})}^{\frac{2}{3}}$

Step 3

Cancel the common factor of $32$ and $4$ .

Factor $4$ out of $32 x^{6} y^{4}$ .

${(\frac{4 (8 x^{6} y^{4})}{4 y})}^{\frac{2}{3}}$

Cancel the common factors.

Factor $4$ out of $4 y$ .

${(\frac{4 (8 x^{6} y^{4})}{4 (y)})}^{\frac{2}{3}}$

Cancel the common factor.

${(\frac{4 (8 x^{6} y^{4})}{4 y})}^{\frac{2}{3}}$

Rewrite the expression.

${(\frac{8 x^{6} y^{4}}{y})}^{\frac{2}{3}}$

Step 4

Cancel the common factor of $y^{4}$ and $y$ .

Factor $y$ out of $8 x^{6} y^{4}$ .

${(\frac{y (8 x^{6} y^{3})}{y})}^{\frac{2}{3}}$

Cancel the common factors.

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

${(\frac{y (8 x^{6} y^{3})}{y^{1}})}^{\frac{2}{3}}$

Factor $y$ out of $y^{1}$ .

${(\frac{y (8 x^{6} y^{3})}{y \cdot 1})}^{\frac{2}{3}}$

Cancel the common factor.

${(\frac{y (8 x^{6} y^{3})}{y \cdot 1})}^{\frac{2}{3}}$

Rewrite the expression.

${(\frac{8 x^{6} y^{3}}{1})}^{\frac{2}{3}}$

Divide $8 x^{6} y^{3}$ by $1$ .

${(8 x^{6} y^{3})}^{\frac{2}{3}}$

Step 5

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

Apply the product rule to $8 x^{6} y^{3}$ .

${(8 x^{6})}^{\frac{2}{3}} {(y^{3})}^{\frac{2}{3}}$

Apply the product rule to $8 x^{6}$ .

$8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}} {(y^{3})}^{\frac{2}{3}}$

Step 6

Simplify the expression.

Rewrite $8$ as $2^{3}$ .

${(2^{3})}^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}} {(y^{3})}^{\frac{2}{3}}$

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

$2^{3 (\frac{2}{3})} {(x^{6})}^{\frac{2}{3}} {(y^{3})}^{\frac{2}{3}}$

Step 7

Cancel the common factor of $3$ .

Cancel the common factor.

$2^{3 (\frac{2}{3})} {(x^{6})}^{\frac{2}{3}} {(y^{3})}^{\frac{2}{3}}$

Rewrite the expression.

$2^{2} {(x^{6})}^{\frac{2}{3}} {(y^{3})}^{\frac{2}{3}}$

Step 8

Simplify the expression.

Raise $2$ to the power of $2$ .

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

Multiply the exponents in ${(x^{6})}^{\frac{2}{3}}$ .

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $2$ .

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x^{4} y^{2}$

Simplify ((32x^4y^4)/(4x^-2y))^(2/3)

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