Simplify (a^-4b^-1c)^-2(a^2bc)^2

${(a^{- 4} b^{- 1} c)}^{- 2} {(a^{2} b c)}^{2}$

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

${(\frac{1}{a^{4}} b^{- 1} c)}^{- 2} {(a^{2} b c)}^{2}$

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

${(\frac{1}{a^{4}} \cdot \frac{1}{b} c)}^{- 2} {(a^{2} b c)}^{2}$

Combine.

${(\frac{1 \cdot 1}{a^{4} b} c)}^{- 2} {(a^{2} b c)}^{2}$

Multiply $1$ by $1$ .

${(\frac{1}{a^{4} b} c)}^{- 2} {(a^{2} b c)}^{2}$

Combine $\frac{1}{a^{4} b}$ and $c$ .

${(\frac{c}{a^{4} b})}^{- 2} {(a^{2} b c)}^{2}$

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{a^{4} b}{c})}^{2} {(a^{2} b c)}^{2}$

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

Apply the product rule to $\frac{a^{4} b}{c}$ .

$\frac{{(a^{4} b)}^{2}}{c^{2}} {(a^{2} b c)}^{2}$

Apply the product rule to $a^{4} b$ .

$\frac{{(a^{4})}^{2} b^{2}}{c^{2}} {(a^{2} b c)}^{2}$

Multiply the exponents in ${(a^{4})}^{2}$ .

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

$\frac{a^{4 \cdot 2} b^{2}}{c^{2}} {(a^{2} b c)}^{2}$

Multiply $4$ by $2$ .

$\frac{a^{8} b^{2}}{c^{2}} {(a^{2} b c)}^{2}$

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

Apply the product rule to $a^{2} b c$ .

$\frac{a^{8} b^{2}}{c^{2}} ({(a^{2} b)}^{2} c^{2})$

Apply the product rule to $a^{2} b$ .

$\frac{a^{8} b^{2}}{c^{2}} ({(a^{2})}^{2} b^{2} c^{2})$

Cancel the common factor of $c^{2}$ .

Factor $c^{2}$ out of ${(a^{2})}^{2} b^{2} c^{2}$ .

$\frac{a^{8} b^{2}}{c^{2}} (c^{2} ({(a^{2})}^{2} b^{2}))$

Cancel the common factor.

$\frac{a^{8} b^{2}}{c^{2}} (c^{2} ({(a^{2})}^{2} b^{2}))$

Rewrite the expression.

$a^{8} b^{2} ({(a^{2})}^{2} b^{2})$

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

$a^{8} b^{2 + 2} {(a^{2})}^{2}$

Add $2$ and $2$ .

$a^{8} b^{4} {(a^{2})}^{2}$

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

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

$a^{8} b^{4} a^{2 \cdot 2}$

Multiply $2$ by $2$ .

$a^{8} b^{4} a^{4}$

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

Move $a^{4}$ .

$a^{4} a^{8} b^{4}$

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

$a^{4 + 8} b^{4}$

Add $4$ and $8$ .

$a^{12} b^{4}$

Simplify (a^-4b^-1c)^-2(a^2bc)^2

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