Question # The values of x that satisfy the equation with rational exponents displaystyle{left({x}+{5}right)}^{{frac{3}{{2}}}}={8} and check all the proposed solutions.

ANSWERED The values of x that satisfy the equation with rational exponents $$\displaystyle{\left({x}+{5}\right)}^{{\frac{3}{{2}}}}={8}$$ and check all the proposed solutions. 2020-11-09
To isolate the variable, raise both sides of the equation to the $$\displaystyle{\left(\frac{2}{{3}}\right)}\ \text{power because}\ \displaystyle{\left(\frac{2}{{3}}\right)}\ \text{is the reciprocal of}\ \displaystyle{\left(\frac{3}{{2}}\right)}:$$
$$\displaystyle{\left({\left({x}+{5}\right)}^{{\frac{2}{{3}}}}\right)}^{{\frac{2}{{3}}}}={\left({8}\right)}^{{\frac{2}{{3}}}}$$
Simplify it further:
$$\displaystyle{x}+{5}={\left({2}^{3}\right)}^{{\frac{2}{{3}}}}$$
$$\displaystyle{x}+{5}={\left({2}\right)}^{{{3}\cdot\frac{2}{{3}}}}$$
$$\displaystyle{x}+{5}={\left({2}\right)}^{2}$$
$$\displaystyle{x}+{5}={4}$$
Subtract 5 from both sides of the equation:
$$\displaystyle{x}+{5}-{5}={4}-{5}$$
Simplify further:
$$x =\ -1$$
Therefore, $$x =\ -1\ \text{is the}\ \displaystyle{\sqrt[]{}}$$ of the equation.
Check:
Subtitute $$x =\ -1$$ into the original equation:
$$\displaystyle{\left({\left(-{1}+{5}\right)}\right)}^{{\frac{3}{{2}}}}={8}.$$
Simplify further:
$$\displaystyle{\left({4}\right)}^{{\frac{3}{{2}}}}={8}$$
$$\displaystyle{\left({2}^{2}\right)}^{{\frac{3}{{2}}}}={8}$$
$$\displaystyle{\left({2}\right)}^{3}={8}$$
$$8 = 8$$
Thus, the left-hand side is equal to the right-hand side of the original expression.
Conclusion:
Hence, $$\displaystyle{x}={\left\lbrace-{1}\right\rbrace}\ \text{is the solution set of the equation}\ \displaystyle{\left({x}+{5}\right)}^{{\frac{3}{{2}}}}={8}$$ is verified