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.

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.