Simplify (12x^8y^-7)/((4x^-2y^-6)^2)

$\frac{12 x^{8} y^{- 7}}{{(4 x^{- 2} y^{- 6})}^{2}}$

Move $y^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12 x^{8}}{{(4 x^{- 2} y^{- 6})}^{2} y^{7}}$

Simplify the denominator.

Apply the product rule to $4 x^{- 2} y^{- 6}$ .

$\frac{12 x^{8}}{{(4 x^{- 2})}^{2} {(y^{- 6})}^{2} y^{7}}$

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

$\frac{12 x^{8}}{{(4 \frac{1}{x^{2}})}^{2} {(y^{- 6})}^{2} y^{7}}$

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

$\frac{12 x^{8}}{{(\frac{4}{x^{2}})}^{2} {(y^{- 6})}^{2} y^{7}}$

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

$\frac{12 x^{8}}{\frac{4^{2}}{{(x^{2})}^{2}} {(y^{- 6})}^{2} y^{7}}$

Raise $4$ to the power of $2$ .

$\frac{12 x^{8}}{\frac{16}{{(x^{2})}^{2}} {(y^{- 6})}^{2} y^{7}}$

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

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

$\frac{12 x^{8}}{\frac{16}{x^{2 \cdot 2}} {(y^{- 6})}^{2} y^{7}}$

Multiply $2$ by $2$ .

$\frac{12 x^{8}}{\frac{16}{x^{4}} {(y^{- 6})}^{2} y^{7}}$

Multiply the exponents in ${(y^{- 6})}^{2}$ .

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

$\frac{12 x^{8}}{\frac{16}{x^{4}} y^{- 6 \cdot 2} y^{7}}$

Multiply $- 6$ by $2$ .

$\frac{12 x^{8}}{\frac{16}{x^{4}} y^{- 12} y^{7}}$

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

$\frac{12 x^{8}}{\frac{16}{x^{4}} \cdot \frac{1}{y^{12}} y^{7}}$

Combine exponents.

Multiply $\frac{16}{x^{4}}$ by $\frac{1}{y^{12}}$ .

$\frac{12 x^{8}}{\frac{16}{x^{4} y^{12}} y^{7}}$

Combine $\frac{16}{x^{4} y^{12}}$ and $y^{7}$ .

$\frac{12 x^{8}}{\frac{16 y^{7}}{x^{4} y^{12}}}$

Reduce the expression $\frac{16 y^{7}}{x^{4} y^{12}}$ by cancelling the common factors.

Factor $y^{7}$ out of $16 y^{7}$ .

$\frac{12 x^{8}}{\frac{y^{7} \cdot 16}{x^{4} y^{12}}}$

Factor $y^{7}$ out of $x^{4} y^{12}$ .

$\frac{12 x^{8}}{\frac{y^{7} \cdot 16}{y^{7} (x^{4} y^{5})}}$

Cancel the common factor.

$\frac{12 x^{8}}{\frac{y^{7} \cdot 16}{y^{7} (x^{4} y^{5})}}$

Rewrite the expression.

$\frac{12 x^{8}}{\frac{16}{x^{4} y^{5}}}$

Multiply the numerator by the reciprocal of the denominator.

$12 x^{8} \frac{x^{4} y^{5}}{16}$

Cancel the common factor of $4$ .

Factor $4$ out of $12 x^{8}$ .

$4 (3 x^{8}) \frac{x^{4} y^{5}}{16}$

Factor $4$ out of $16$ .

$4 (3 x^{8}) \frac{x^{4} y^{5}}{4 (4)}$

Cancel the common factor.

$4 (3 x^{8}) \frac{x^{4} y^{5}}{4 \cdot 4}$

Rewrite the expression.

$3 x^{8} \frac{x^{4} y^{5}}{4}$

Combine $3$ and $\frac{x^{4} y^{5}}{4}$ .

$x^{8} \frac{3 (x^{4} y^{5})}{4}$

Combine $x^{8}$ and $\frac{3 (x^{4} y^{5})}{4}$ .

$\frac{x^{8} (3 (x^{4} y^{5}))}{4}$

Multiply $x^{8}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$\frac{x^{4} x^{8} (3 (y^{5}))}{4}$

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

$\frac{x^{4 + 8} (3 (y^{5}))}{4}$

Add $4$ and $8$ .

$\frac{x^{12} (3 (y^{5}))}{4}$

Move $3$ to the left of $x^{12}$ .

$\frac{3 x^{12} y^{5}}{4}$

Simplify (12x^8y^-7)/((4x^-2y^-6)^2)

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