Solve the Rational Equation for x 2x^(2/3)=8

$2 x^{\frac{2}{3}} = 8$

Step 1

Divide each term in $2 x^{\frac{2}{3}} = 8$ by $2$ and simplify.

Divide each term in $2 x^{\frac{2}{3}} = 8$ by $2$ .

$\frac{2 x^{\frac{2}{3}}}{2} = \frac{8}{2}$

Simplify the left side.

Cancel the common factor.

$\frac{2 x^{\frac{2}{3}}}{2} = \frac{8}{2}$

Divide $x^{\frac{2}{3}}$ by $1$ .

$x^{\frac{2}{3}} = \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

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

Step 2

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

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

Step 3

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Simplify the expression.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = \pm 2^{3}$

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

$x = \pm 8$

Step 4

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.

$x = 8$

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

$x = - 8$

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

$x = 8, - 8$

Solve the Rational Equation for x 2x^(2/3)=8

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