Question

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

Rational exponents and radicals
ANSWERED
asked 2020-11-10
The values of x that satisfy the equation with rational exponents \(\displaystyle{x}^{{\frac{3}{{2}}}}={8}\) and check all the proposed solutions.

Answers (1)

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.
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