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

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{5}{2} \cdot \frac{2}{5}} = 32^{\frac{2}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{5}{2} \cdot \frac{2}{5}} = 32^{\frac{2}{5}}$

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Simplify the expression.

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 2^{2}$

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

$x = 4$