Question # The values of x that satisfy the equation with rational exponents displaystyle{x}^{{frac{3}{{2}}}}={8} and check all the proposed solutions.

ANSWERED The values of x that satisfy the equation with rational exponents $$\displaystyle{x}^{{\frac{3}{{2}}}}={8}$$ and check all the proposed solutions. 2020-11-11
Given:
The equation with rational exponents $$\displaystyle{x}^{{\frac{3}{{2}}}}={8}.$$
To isolate the variable, raise both sides of the equation to the $$\displaystyle{\left(\frac{2}{{3}}\right)}\ \text{power because}\ \displaystyle{\left(\frac{2}{{3}}\right)}\ \text{is reciprocal of}\ \displaystyle{\left(\frac{3}{{2}}\right)}:$$
$$\displaystyle{\left({x}^{{\frac{3}{{2}}}}\right)}^{{\frac{2}{{3}}}}={\left({8}\right)}^{{\frac{2}{{3}}}}$$
Simplify it further:
$$\displaystyle{x}={\left({2}^{3}\right)}^{{\frac{2}{{3}}}}$$
$$\displaystyle{x}={\left({2}\right)}^{{{3}\times\frac{2}{{3}}}}$$
$$\displaystyle{x}={\left({2}\right)}^{2}$$
$$x = 4$$
Therefore, $$x = 4\ \text{is the}\ \displaystyle{\sqrt[]{}}$$ of the equation.
Check:
Substitute $$x = 4$$ into the original equation:
$$\displaystyle{\left({4}\right)}^{{\frac{3}{{2}}}}={8}$$
Simplify further:
$$\displaystyle{\left({2}^{2}\right)}^{{\frac{3}{{2}}}}={8}$$
$$\displaystyle{\left({2}\right)}^{{{2}\times\frac{3}{{2}}}}={8}$$
$$\displaystyle{\left({2}\right)}^{3}={8}$$
$$8 = 8$$
Thus, left-hand side is equal to the right-hand side of the original expression.
Conclusion:
Hence, $$\displaystyle{x}={\left\lbrace{4}\right\rbrace}\ \text{is the solution set of the equation}\ \displaystyle{x}^{{\frac{3}{{2}}}}={8}$$ and is verified.