Simplify ((a^-2b^-1c^-4)/(a^4b^-3c^0))^-3

${(\frac{a^{- 2} b^{- 1} c^{- 4}}{a^{4} b^{- 3} c^{0}})}^{- 3}$

Step 1

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

${(\frac{b^{- 1} c^{- 4}}{a^{4} b^{- 3} c^{0} a^{2}})}^{- 3}$

Step 2

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

${(\frac{b^{- 1}}{a^{4} b^{- 3} c^{0} a^{2} c^{4}})}^{- 3}$

Step 3

Simplify the denominator.

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

Move $a^{2}$ .

${(\frac{b^{- 1}}{a^{2} a^{4} b^{- 3} c^{0} c^{4}})}^{- 3}$

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

${(\frac{b^{- 1}}{a^{2 + 4} b^{- 3} c^{0} c^{4}})}^{- 3}$

Add $2$ and $4$ .

${(\frac{b^{- 1}}{a^{6} b^{- 3} c^{0} c^{4}})}^{- 3}$

Simplify $a^{6} b^{- 3} c^{0}$ .

${(\frac{b^{- 1}}{a^{6} b^{- 3} c^{4}})}^{- 3}$

Step 4

Move $b^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{b^{- 1} b^{3}}{a^{6} c^{4}})}^{- 3}$

Step 5

Multiply $b^{- 1}$ by $b^{3}$ by adding the exponents.

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

${(\frac{b^{- 1 + 3}}{a^{6} c^{4}})}^{- 3}$

Add $- 1$ and $3$ .

${(\frac{b^{2}}{a^{6} c^{4}})}^{- 3}$

Step 6

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

${(\frac{a^{6} c^{4}}{b^{2}})}^{3}$

Step 7

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

Apply the product rule to $\frac{a^{6} c^{4}}{b^{2}}$ .

$\frac{{(a^{6} c^{4})}^{3}}{{(b^{2})}^{3}}$

Apply the product rule to $a^{6} c^{4}$ .

$\frac{{(a^{6})}^{3} {(c^{4})}^{3}}{{(b^{2})}^{3}}$

Step 8

Simplify the numerator.

Multiply the exponents in ${(a^{6})}^{3}$ .

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

$\frac{a^{6 \cdot 3} {(c^{4})}^{3}}{{(b^{2})}^{3}}$

Multiply $6$ by $3$ .

$\frac{a^{18} {(c^{4})}^{3}}{{(b^{2})}^{3}}$

Multiply the exponents in ${(c^{4})}^{3}$ .

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

$\frac{a^{18} c^{4 \cdot 3}}{{(b^{2})}^{3}}$

Multiply $4$ by $3$ .

$\frac{a^{18} c^{12}}{{(b^{2})}^{3}}$

Step 9

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

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

$\frac{a^{18} c^{12}}{b^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{a^{18} c^{12}}{b^{6}}$

Simplify ((a^-2b^-1c^-4)/(a^4b^-3c^0))^-3

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