Question

In the following exercises,solve each eqution.

$2^{x^2}=16$

Short answer

The solution is$x=-2andx=2.$

Step-by-step solution

Step 1.Given information

Consider the given quadratic equation $2^{x^2}=16$.

Solve the equation for x as follows;

Step 2.First write both sides of the equation with the same base

Because left side of the equation has base 2,so write the right side of the equation with base 3 as

$2^{x^2}=2^4[2^4=16]$

Because the base are same 2,so the exponent must be equal.

$x^2=4$.

Step 3.Now solve the equation

$$\begin{gathered} x=\pm \sqrt{4}[squarerootproperty] \\ x=\pm 2 \end{gathered}$$

So,the values of$x=-2andx=2$

Step 4.Check

$$\begin{gathered} 2^{x^2}=162^{x^2}=16 \\ {2}^{{(-2)}^2}=162^{{\left(2\right)}^2}=16 \\ {2}^4=162^4=16 \\ 16=16\left[true\right]16=16\left[true\right] \end{gathered}$$

Therfore,the solution are$x=-2andx=2$