Simplify ((3a^-4)/(b^7))^3

${(\frac{3 a^{- 4}}{b^{7}})}^{3}$

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

${(\frac{3}{b^{7} a^{4}})}^{3}$

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

Apply the product rule to $\frac{3}{b^{7} a^{4}}$ .

$\frac{3^{3}}{{(b^{7} a^{4})}^{3}}$

Apply the product rule to $b^{7} a^{4}$ .

$\frac{3^{3}}{{(b^{7})}^{3} {(a^{4})}^{3}}$

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

$\frac{27}{{(b^{7})}^{3} {(a^{4})}^{3}}$

Simplify the denominator.

Multiply the exponents in ${(b^{7})}^{3}$ .

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

$\frac{27}{b^{7 \cdot 3} {(a^{4})}^{3}}$

Multiply $7$ by $3$ .

$\frac{27}{b^{21} {(a^{4})}^{3}}$

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

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

$\frac{27}{b^{21} a^{4 \cdot 3}}$

Multiply $4$ by $3$ .

$\frac{27}{b^{21} a^{12}}$