Chapter 7.4, Problem 47PPE

To determine

To simplify: the expression ${\left(\frac{b^{\frac{4}{5}}}{b^7}\right)}^{-5}$

Answer to Problem 47PPE

The required answer is: $b^{31}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{b^{\frac{4}{5}}}{b^7}\right)}^{-5}$

Calculation:

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

(Rewrite using positive exponents)

  $=\frac{1}{\left(\frac{{(b^{\frac{4}{5}})}^5}{{(b^7)}^5}\right)}$

(Raise the quotient to fifth power)

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

(Multiply the exponents in the quotient)

  $=1\cdot \frac{{(b)}^{35}}{{(b)}^4}$

(Multiply by the reciprocal of $\frac{{(b)}^4}{{(b)}^{35}}$ which is $\frac{{(b)}^{35}}{{(b)}^4}$ )

  $=b^{35-4}$

(Subtract exponents when dividing powers with the same base)

  $=b^{31}$

Simplify)