Solve the Rational Equation for x x^(5/2)=243

$x^{\frac{5}{2}} = 243$

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

${(x^{\frac{5}{2}})}^{\frac{2}{5}} = 243^{\frac{2}{5}}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{5}{2} \cdot \frac{2}{5}} = 243^{\frac{2}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{5}{2} \cdot \frac{2}{5}} = 243^{\frac{2}{5}}$

Rewrite the expression.

$x^{\frac{1}{2} \cdot 2} = 243^{\frac{2}{5}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 243^{\frac{2}{5}}$

Rewrite the expression.

$x^{1} = 243^{\frac{2}{5}}$

Simplify.

$x = 243^{\frac{2}{5}}$

Simplify the right side.

Simplify $243^{\frac{2}{5}}$ .

Simplify the expression.

Rewrite $243$ as $3^{5}$ .

$x = {(3^{5})}^{\frac{2}{5}}$

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

$x = 3^{5 (\frac{2}{5})}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x = 3^{5 (\frac{2}{5})}$

Rewrite the expression.

$x = 3^{2}$

Raise $3$ to the power of $2$ .

$x = 9$