Solve the Rational Equation for x (x-1)^(5/2)=32

${(x - 1)}^{\frac{5}{2}} = 32$

Step 1

Raise each side of the equation to the power of $\frac{2}{5}$ to eliminate the fractional exponent on the left side.

${({(x - 1)}^{\frac{5}{2}})}^{\frac{2}{5}} = 32^{\frac{2}{5}}$

Step 2

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 1)}^{\frac{5}{2}})}^{\frac{2}{5}}$ .

Multiply the exponents in ${({(x - 1)}^{\frac{5}{2}})}^{\frac{2}{5}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 1)}^{\frac{5}{2} \cdot \frac{2}{5}} = 32^{\frac{2}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 1)}^{\frac{5}{2} \cdot \frac{2}{5}} = 32^{\frac{2}{5}}$

Rewrite the expression.

${(x - 1)}^{\frac{1}{2} \cdot 2} = 32^{\frac{2}{5}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} = 32^{\frac{2}{5}}$

Rewrite the expression.

${(x - 1)}^{1} = 32^{\frac{2}{5}}$

Simplify.

$x - 1 = 32^{\frac{2}{5}}$

Simplify the right side.

Simplify $32^{\frac{2}{5}}$ .

Simplify the expression.

Rewrite $32$ as $2^{5}$ .

$x - 1 = {(2^{5})}^{\frac{2}{5}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x - 1 = 2^{5 (\frac{2}{5})}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x - 1 = 2^{5 (\frac{2}{5})}$

Rewrite the expression.

$x - 1 = 2^{2}$

Raise $2$ to the power of $2$ .

$x - 1 = 4$

Step 3

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

$x = 4 + 1$

Add $4$ and $1$ .

$x = 5$