Simplify ((4x^-2y^2)/(12x^-4y^-1))^3

${(\frac{4 x^{- 2} y^{2}}{12 x^{- 4} y^{- 1}})}^{3}$

Step 1

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

${(\frac{4 x^{- 2} y^{2} x^{4}}{12 y^{- 1}})}^{3}$

Step 2

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

${(\frac{4 x^{- 2} y^{2} x^{4} y}{12})}^{3}$

Step 3

Simplify the numerator.

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

Move $x^{4}$ .

${(\frac{4 (x^{4} x^{- 2}) y^{2} y}{12})}^{3}$

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

${(\frac{4 x^{4 - 2} y^{2} y}{12})}^{3}$

Subtract $2$ from $4$ .

${(\frac{4 x^{2} y^{2} y}{12})}^{3}$

Multiply $y^{2}$ by $y$ by adding the exponents.

Move $y$ .

${(\frac{4 x^{2} (y \cdot y^{2})}{12})}^{3}$

Multiply $y$ by $y^{2}$ .

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

${(\frac{4 x^{2} (y^{1} y^{2})}{12})}^{3}$

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

${(\frac{4 x^{2} y^{1 + 2}}{12})}^{3}$

Add $1$ and $2$ .

${(\frac{4 x^{2} y^{3}}{12})}^{3}$

Step 4

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4 x^{2} y^{3}$ .

${(\frac{4 (x^{2} y^{3})}{12})}^{3}$

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

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

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

Apply the product rule to $x^{2} y^{3}$ .

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

Step 6

Simplify the numerator.

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

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

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

Multiply $2$ by $3$ .

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

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

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

$\frac{x^{6} y^{3 \cdot 3}}{3^{3}}$

Multiply $3$ by $3$ .

$\frac{x^{6} y^{9}}{3^{3}}$

Step 7

Raise $3$ to the power of $3$ .

$\frac{x^{6} y^{9}}{27}$

Simplify ((4x^-2y^2)/(12x^-4y^-1))^3

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