Chapter 7.3, Problem 83PPS

To determine

To solve the given equation.

Answer to Problem 83PPS

  $x=-\frac{3}{2}$

Explanation of Solution

Given:

Equation: ${25}^x=\frac{1}{125}$

Formula used:

Exponent rules:

  ${(a^b)}^c=a^{bc}$

  $a^{f(x)}=a^{g(x)},$ then $f(x)=g(x)$

Calculation:

Consider the given equation,

  $\Rightarrow {25}^x=\frac{1}{125}$

Here, 25 can be written as $5^2$ and $\frac{1}{125}$ can be written as ${125}^{-1}$ . Further 125 can be written as $5^3.$

Hence,

  $\begin{array}{c}$ {(5^2)}^x={125}^{-1}$$ {(5^2)}^x={(5^3)}^{-1}$$ 5^2{}^x=5^{-3}[\because {(a^b)}^c=a^{bc}]$2x=-3[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $2x=-3$

Dividing 2 on both sides and simplifying,

  $\begin{array}{c}\frac{2x}{2}=\frac{-3}{2}\\ x=-\frac{3}{2}\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=-\frac{3}{2}.$