Question

In the following exercises, solve each equation:

$4^{x^2}=4$

Short answer

The solutions of the given equation are$x=-1andx=1.$

Step-by-step solution

Step 1. Given Information 

The given equation is$4^{x^2}=4.$

We have to solve it.

Step 2. Solve the equation  

To solve the equation, write both sides of the equation with the same base.

So,$4^{x^2}=4^1.$

Now, write a new equation by setting the exponents equal.

$$\begin{gathered} x^2=1 \\ x=\pm \sqrt{1} \\ x=\pm 1 \end{gathered}$$

Step 3. Verify the solution 

To verify the solution,

Substitute $x=1$in the original equation

$$\begin{gathered} 4^{{\left(1\right)}^2}=4 \\ {4}^1=4 \\ 4=4 \end{gathered}$$

It is true.

Now, substitute $x=-1$in the original equation

$$\begin{gathered} 4^{{\left(-1\right)}^2}=4 \\ {4}^1=4 \\ 4=4 \end{gathered}$$

Hence, the solutions are$x=1,-1.$