Chapter 7.2, Problem 14PPS

To determine

To calculate: The value of y in the equation $8^{2y+4}={16}^{y+1}$ .

Answer to Problem 14PPS

The value of y in the equation $8^{2y+4}={16}^{y+1}$ is $-4$ .

Explanation of Solution

Given information:

The given equation is $8^{2y+4}={16}^{y+1}$ .

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,

  $8^{2y+4}={16}^{y+1}$

Since 8 can be written as $2^3$ and 16 can be written as $2^4$ .So, rewrite the above equation as like this,

  $\begin{array}{c}{8}^{2y+4}={16}^{y+1}\\ 2^{3\left(2y+4\right)}=2^{4\left(y+1\right)}\end{array}$

Now, apply the property of equality of exponential functions:

  $\begin{array}{c}3\left(2y+4\right)=4\left(y+1\right)\\ 6y+12=4y+4\\ 6y-4y=4-12\\ 2y=-8\end{array}$

Proceed to next,

  $\begin{array}{c}2y=-8\\ y=\frac{-8}{2}\\ y=-4\end{array}$

Therefore, the value of y in the equation $8^{2y+4}={16}^{y+1}$ is $-4$ .