Find All Complex Solutions x^(2/7)=2

$x^{\frac{2}{7}} = 2$

Step 1

Raise each side of the equation to the power of $\frac{7}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{7}})}^{\frac{7}{2}} = \pm 2^{\frac{7}{2}}$

Step 2

Simplify the left side.

Simplify ${(x^{\frac{2}{7}})}^{\frac{7}{2}}$ .

Multiply the exponents in ${(x^{\frac{2}{7}})}^{\frac{7}{2}}$ .

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

$x^{\frac{2}{7} \cdot \frac{7}{2}} = \pm 2^{\frac{7}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{2}{7} \cdot \frac{7}{2}} = \pm 2^{\frac{7}{2}}$

Rewrite the expression.

$x^{\frac{1}{7} \cdot 7} = \pm 2^{\frac{7}{2}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = \pm 2^{\frac{7}{2}}$

Rewrite the expression.

$x^{1} = \pm 2^{\frac{7}{2}}$

Simplify.

$x = \pm 2^{\frac{7}{2}}$

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.

$x = 2^{\frac{7}{2}}$

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

$x = - 2^{\frac{7}{2}}$

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

$x = 2^{\frac{7}{2}}, - 2^{\frac{7}{2}}$

Step 4

Exclude the solutions that do not make $x^{\frac{2}{7}} = 2$ true.

$x = 2^{\frac{7}{2}}$