Chapter 7, Problem 12CR

To determine

The value of the exponent.

Answer to Problem 12CR

The value of the exponent is -9.

Explanation of Solution

Given information:An equation given with a missing exponent. The equation given is ${\left(\frac{a^{5}b{c}^3}{b^4{c}^2}\right)}^{?}=\frac{b^{27}}{a^{45}{c}^9}$ .

Calculation:

Let, the missing exponent be x.

It can be written as,

  $\begin{array}{l}{\left(\frac{a^{5}b{c}^3}{b^4{c}^2}\right)}^x=\frac{b^{27}}{a^{45}{c}^9}\\ \Rightarrow {\left(a^5{b}^{(1-4)}{c}^{(3-2)}\right)}^x=a^{-45}{b}^{27}{c}^{-9}\\ \Rightarrow {\left(a^5{b}^{-3}c\right)}^x=a^{-45}{b}^{27}{c}^{-9}\\ \Rightarrow a^{5x}{b}^{-3x}{c}^x=a^{-45}{b}^{27}{c}^{-9}\end{array}$

By equating the co-efficient of a, we can write,

  $\begin{array}{l}5x=-45\\ \Rightarrow x=\frac{-45}{5}\\ \Rightarrow x=-9\end{array}$

So, the value of the exponent = -9.