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

${(a^{2} b^{- 4})}^{3} {(c^{3} d)}^{- 1}$

Step 1

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

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

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

Multiply $4$ by $3$ .

$\frac{a^{6}}{b^{12}} {(c^{3} d)}^{- 1}$

Step 4

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

$\frac{a^{6}}{b^{12}} \cdot \frac{1}{c^{3} d}$

Step 5

Combine.

$\frac{a^{6} \cdot 1}{b^{12} (c^{3} d)}$

Step 6

Multiply $a^{6}$ by $1$ .

$\frac{a^{6}}{b^{12} c^{3} d}$