Simplify ((b^(4/5))/(b^7))^-5

${(\frac{b^{\frac{4}{5}}}{b^{7}})}^{- 5}$

Move $b^{\frac{4}{5}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

${(\frac{1}{b^{7} b^{- \frac{4}{5}}})}^{- 5}$

Multiply $b^{7}$ by $b^{- \frac{4}{5}}$ by adding the exponents.

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

${(\frac{1}{b^{7 - \frac{4}{5}}})}^{- 5}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(\frac{1}{b^{7 \cdot \frac{5}{5} - \frac{4}{5}}})}^{- 5}$

Combine $7$ and $\frac{5}{5}$ .

${(\frac{1}{b^{\frac{7 \cdot 5}{5} - \frac{4}{5}}})}^{- 5}$

Combine the numerators over the common denominator.

${(\frac{1}{b^{\frac{7 \cdot 5 - 4}{5}}})}^{- 5}$

Simplify the numerator.

Multiply $7$ by $5$ .

${(\frac{1}{b^{\frac{35 - 4}{5}}})}^{- 5}$

Subtract $4$ from $35$ .

${(\frac{1}{b^{\frac{31}{5}}})}^{- 5}$

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

${(b^{\frac{31}{5}})}^{5}$

Multiply the exponents in ${(b^{\frac{31}{5}})}^{5}$ .

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

$b^{\frac{31}{5} \cdot 5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$b^{\frac{31}{5} \cdot 5}$

Rewrite the expression.

$b^{31}$