Solve by Factoring x^(5/3)=243

$x^{\frac{5}{3}} = 243$

Step 1

Subtract $243$ from both sides of the equation.

$x^{\frac{5}{3}} - 243 = 0$

Step 2

Add $243$ to both sides of the equation.

$x^{\frac{5}{3}} = 243$

Step 3

Raise each side of the equation to the power of $\frac{3}{5}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{5}{3}})}^{\frac{3}{5}} = 243^{\frac{3}{5}}$

Step 4

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

Multiply the exponents in ${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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

$x^{\frac{5}{3} \cdot \frac{3}{5}} = 243^{\frac{3}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{5}{3} \cdot \frac{3}{5}} = 243^{\frac{3}{5}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = 243^{\frac{3}{5}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 243^{\frac{3}{5}}$

Rewrite the expression.

$x^{1} = 243^{\frac{3}{5}}$

Simplify.

$x = 243^{\frac{3}{5}}$

Simplify the right side.

Simplify $243^{\frac{3}{5}}$ .

Simplify the expression.

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 3^{3}$

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

$x = 27$