Question

solve the equation with rational exponents. (x - 1)^(5/2)=32 select the correct choice below and, if necessary, fill in the answer box to complete your o a. the solution set is { }. (simplify your answer. use a comma to separate answers as needed.) o b. the solution set is the empty set.

Explanation:

Step1: Isolate the base - exponent form

Raise both sides to the $\frac{2}{5}$ power. The left - hand side becomes $((x - 1)^{\frac{5}{2}})^{\frac{2}{5}}$ and the right - hand side becomes $32^{\frac{2}{5}}$. According to the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, so $((x - 1)^{\frac{5}{2}})^{\frac{2}{5}}=x - 1$. And $32^{\frac{2}{5}}=(2^{5})^{\frac{2}{5}}$.

Step2: Simplify the right - hand side

Using the power - of - a - power rule again, $(2^{5})^{\frac{2}{5}}=2^{5\times\frac{2}{5}} = 2^{2}=4$. So we have the equation $x−1 = 4$.

Step3: Solve for x

Add 1 to both sides of the equation $x−1 = 4$. We get $x=4 + 1=5$.

Answer:

A. The solution set is $\{5\}$