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

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

Step 1

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

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

Step 2

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

Move $x^{2}$ .

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

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

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

Add $2$ and $3$ .

${(\frac{12 x^{5} y^{- 4}}{4 y^{- 4}})}^{- 2}$

Step 3

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

Factor $4$ out of $12 x^{5} y^{- 4}$ .

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

Cancel the common factors.

Factor $4$ out of $4 y^{- 4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

Cancel the common factor.

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

Divide $3 x^{5}$ by $1$ .

${(3 x^{5})}^{- 2}$

Step 5

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

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

Step 6

Simplify the denominator.

Apply the product rule to $3 x^{5}$ .

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

Raise $3$ to the power of $2$ .

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

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$\frac{1}{9 x^{5 \cdot 2}}$

Multiply $5$ by $2$ .

$\frac{1}{9 x^{10}}$