Solve the Rational Equation for x (x-2)^(2/3)=4

${(x - 2)}^{\frac{2}{3}} = 4$

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

${({(x - 2)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Simplify the exponent.

Simplify the left side.

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

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

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

${(x - 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Rewrite the expression.

${(x - 2)}^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Rewrite the expression.

${(x - 2)}^{1} = \pm 4^{\frac{3}{2}}$

Simplify.

$x - 2 = \pm 4^{\frac{3}{2}}$

Simplify the right side.

Simplify $\pm 4^{\frac{3}{2}}$ .

Simplify the expression.

Rewrite $4$ as $2^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$x - 2 = \pm 8$

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.

$x - 2 = 8$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$x = 8 + 2$

Add $8$ and $2$ .

$x = 10$

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

$x - 2 = - 8$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$x = - 8 + 2$

Add $- 8$ and $2$ .

$x = - 6$

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

$x = 10, - 6$

Solve the Rational Equation for x (x-2)^(2/3)=4

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