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.

Rational exponents and radicals
ANSWERED
asked 2020-11-08
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.

Expert Answers (1)

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
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