Q. 69 - Precalculus Enhanced with Graphing Utilities

Question

Solve the equation $3^{x^3}=9^x$

and verify your results using a graphing utility.

Step-by-Step Solution

Answer

Required solution set is $\{0,\sqrt{2},-\sqrt{2}\}$

Step 1. Given information

$3^{x^3}=9^x$

Step 2. Calculations

$$\begin{gathered} 3^{x^3}=9^{x } \\ {3}^{x^3}=3^{2x} if a^u=a^v then u=v \\ {x}^3=2x \\ {x}^3-2x=0 \\ x(x^2-2)=0 \\ x(x-\sqrt{2})(x+\sqrt{2})=0 \\ x=0, \sqrt{2},-\sqrt{2} \end{gathered}$$

Step 3. Graphing solution

Graph $y=3^{x^3}$ and $y=9^x$ intersect at (0,1) , $(\sqrt{2},3^{2\sqrt{2}}) , (-\sqrt{2,},3^{-2\sqrt{2}})$

Hence the required solution set is $\{0,\sqrt{2},-\sqrt{2}\}$.