Question

In the following exercises, solve each equation:

$3^{x^2}=81$

Short answer

The solutions of the given equation are$x=2andx=-2.$

Step-by-step solution

Step 1. Given Information 

The given equation is$3^{x^2}=81.$

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,$3^{x^2}=3^4.$

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

$$\begin{gathered} x^2=4 \\ x=\pm \sqrt{4} \\ x=\pm 2 \end{gathered}$$

Step 3. Verify the solution 

To verify the solution,

Substitute $x=2$in the original equation

role="math" localid="1644924523318" $$\begin{gathered} 3^{{\left(2\right)}^2}=81 \\ {3}^4=81 \\ 81=81 \end{gathered}$$

It is true.

Now, substitute $x=-2$in the original equation

$$\begin{gathered} 3^{{(-2)}^2}=81 \\ {3}^4=81 \\ 81=81 \end{gathered}$$

It is also true.

Hence, the solutions are$x=2,-2.$