Chapter 7.3, Problem 84PPS

To determine

To solve the given equation.

Answer to Problem 84PPS

  $x=-5$

Explanation of Solution

Given:

Equation: $6^{8-x}=\frac{1}{216}$

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,

  $\Rightarrow 6^{8-x}=\frac{1}{216}$

Here, $\frac{1}{216}$ can be written as ${216}^{-1}$ . Further 216 can be written as $6^3.$

Hence,

  $\begin{array}{c}{6}^{8-x}={216}^{-1}\\ $ 6^{8-x}={(6^3)}^{-1}$$ 6^{8-x}=6^{-3}[\because {(a^b)}^c=a^{bc}]$8-x=-3[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $8-x=-3$

Adding 8 on both sides and simplifying,

  $\begin{array}{c}8-x+8=-3+8\\ -x=5\end{array}$

Dividing − 1 on both sides and simplifying,

  $\begin{array}{c}\frac{-x}{-1}=\frac{5}{-1}\\ x=-5\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=-5.$