Chapter 8.2, Problem 34PPS

To determine

To calculate: The value of x in the equation $216={\left(\frac{1}{6}\right)}^{x+3}$ .

Answer to Problem 34PPS

The value of x in the equation $216={\left(\frac{1}{6}\right)}^{x+3}$ is $-6$ .

Explanation of Solution

Given information:

The given equation is $216={\left(\frac{1}{6}\right)}^{x+3}$ .

Formula used:

Property of equality of exponential functions:

If $b^x=b^y$ then $x=y$for all $b>0$ ,and $b\ne 1$ .

Calculation:

Consider the given equation,

  $216={\left(\frac{1}{6}\right)}^{x+3}$

Since 216 can be written as $6^3$ and $\frac{1}{6}$ can be written as $6^{-1}$ .So, rewrite the above equation as like this,

  $\begin{array}{c}{6}^3=6^{-\left(x+3\right)}\\ 6^3=6^{-x-3}\end{array}$

Now, apply the property of equality of exponential functions:

  $\begin{array}{c}3=-x-3\\ 3+3=-x\\ 6=-x\\ x=-6\end{array}$

Therefore, the value of x in the equation $216={\left(\frac{1}{6}\right)}^{x+3}$ is $-6$ .