Question

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places :

${\left(\sqrt[3]{2}\right)}^{2-x}=2^{x^2}$

Short answer

The value of x satisfying the equation is$x=-1orx=\frac{2}{3}$

Step-by-step solution

Given information

We are given an equation${\left(\sqrt[3]{2}\right)}^{2-x}=2^{x^2}$.

Simplify

We get,

$$\begin{gathered} {\left(\sqrt[3]{2}\right)}^{2-x}=2^{x^2} \\ {\left(2^{\frac{1}{3}}\right)}^{2-x}=2^{x2} \\ {2}^{\frac{1}{3}(2-x)}=2^{x^2} \end{gathered}$$

Solve for x

As the bases are same we get,