Chapter 8.2, Problem 12PPS

To determine

To calculate: The value of b in the equation ${256}^{b+2}=4^{2-2b}$ .

Answer to Problem 12PPS

The value of b in the equation ${256}^{b+2}=4^{2-2b}$ is $-1$ .

Explanation of Solution

Given information:

The given equation is ${256}^{b+2}=4^{2-2b}$ .

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,

  ${256}^{b+2}=4^{2-2b}$

Since 256 can be written as $4^4$ . So, rewrite the above equation as like this,

  $\begin{array}{c}{256}^{b+2}=4^{2-2b}\\ 4^{4\left(b+2\right)}=4^{2-2b}\end{array}$

Now, apply the property of equality of exponential functions:

  $\begin{array}{c}4\left(b+2\right)=2-2b\\ 4b+8=2-2b\\ 4b+2b=2-8\\ 6b=-6\end{array}$

Proceed to next,

  $\begin{array}{c}6b=-6\\ b=\frac{-6}{6}\\ b=-1\end{array}$

Therefore, the value of b in the equation ${256}^{b+2}=4^{2-2b}$ is $-1$ .