Find All Complex Solutions (x-5)^(2/3)=2

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

Step 1

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Step 2

Simplify the left side.

Simplify ${({(x - 5)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${({(x - 5)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(x - 5)}^{\frac{1}{3} \cdot 3} = \pm 2^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 5)}^{\frac{1}{3} \cdot 3} = \pm 2^{\frac{3}{2}}$

Rewrite the expression.

${(x - 5)}^{1} = \pm 2^{\frac{3}{2}}$

Simplify.

$x - 5 = \pm 2^{\frac{3}{2}}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x - 5 = 2^{\frac{3}{2}}$

Add $5$ to both sides of the equation.

$x = 2^{\frac{3}{2}} + 5$

Next, use the negative value of the $\pm$ to find the second solution.

$x - 5 = - 2^{\frac{3}{2}}$

Add $5$ to both sides of the equation.

$x = - 2^{\frac{3}{2}} + 5$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2^{\frac{3}{2}} + 5, - 2^{\frac{3}{2}} + 5$