Find All Complex Solutions (x-2)^(4/5)=2

${(x - 2)}^{\frac{4}{5}} = 2$

Step 1

Raise each side of the equation to the power of $\frac{5}{4}$ to eliminate the fractional exponent on the left side.

${({(x - 2)}^{\frac{4}{5}})}^{\frac{5}{4}} = \pm 2^{\frac{5}{4}}$

Step 2

Simplify the left side.

Simplify ${({(x - 2)}^{\frac{4}{5}})}^{\frac{5}{4}}$ .

Multiply the exponents in ${({(x - 2)}^{\frac{4}{5}})}^{\frac{5}{4}}$ .

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

${(x - 2)}^{\frac{4}{5} \cdot \frac{5}{4}} = \pm 2^{\frac{5}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(x - 2)}^{\frac{4}{5} \cdot \frac{5}{4}} = \pm 2^{\frac{5}{4}}$

Rewrite the expression.

${(x - 2)}^{\frac{1}{5} \cdot 5} = \pm 2^{\frac{5}{4}}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{5} \cdot 5} = \pm 2^{\frac{5}{4}}$

Rewrite the expression.

${(x - 2)}^{1} = \pm 2^{\frac{5}{4}}$

Simplify.

$x - 2 = \pm 2^{\frac{5}{4}}$

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.

$x - 2 = 2^{\frac{5}{4}}$

Add $2$ to both sides of the equation.

$x = 2^{\frac{5}{4}} + 2$

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

$x - 2 = - 2^{\frac{5}{4}}$

Add $2$ to both sides of the equation.

$x = - 2^{\frac{5}{4}} + 2$

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

$x = 2^{\frac{5}{4}} + 2, - 2^{\frac{5}{4}} + 2$