Evaluate -8=-3-6 cube root of x^2

$- 8 = - 3 - 6 \sqrt[3]{x^{2}}$

Step 1

Rewrite the equation as $- 3 - 6 \sqrt[3]{x^{2}} = - 8$ .

$- 3 - 6 \sqrt[3]{x^{2}} = - 8$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$- 3 - 6 x^{\frac{2}{3}} = - 8$

Step 3

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

Add $3$ to both sides of the equation.

$- 6 x^{\frac{2}{3}} + 8 = 3$

Subtract $8$ from both sides of the equation.

$- 6 x^{\frac{2}{3}} = 3 - 8$

Subtract $8$ from $3$ .

$- 6 x^{\frac{2}{3}} = - 5$

Step 4

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

${(- 6 x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(- 5)}^{\frac{3}{2}}$

Step 5

Simplify the left side.

Simplify ${(- 6 x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Apply the product rule to $- 6 x^{\frac{2}{3}}$ .

${(- 6)}^{\frac{3}{2}} {(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(- 5)}^{\frac{3}{2}}$

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

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

${(- 6)}^{\frac{3}{2}} x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 5)}^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 6)}^{\frac{3}{2}} x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 5)}^{\frac{3}{2}}$

Rewrite the expression.

${(- 6)}^{\frac{3}{2}} x^{\frac{1}{3} \cdot 3} = \pm {(- 5)}^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 6)}^{\frac{3}{2}} x^{\frac{1}{3} \cdot 3} = \pm {(- 5)}^{\frac{3}{2}}$

Rewrite the expression.

${(- 6)}^{\frac{3}{2}} x^{1} = \pm {(- 5)}^{\frac{3}{2}}$

Simplify.

${(- 6)}^{\frac{3}{2}} x = \pm {(- 5)}^{\frac{3}{2}}$

Reorder factors in ${(- 6)}^{\frac{3}{2}} x$ .

$x {(- 6)}^{\frac{3}{2}} = \pm {(- 5)}^{\frac{3}{2}}$

Step 6

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.

$x {(- 6)}^{\frac{3}{2}} = {(- 5)}^{\frac{3}{2}}$

Divide each term in $x {(- 6)}^{\frac{3}{2}} = {(- 5)}^{\frac{3}{2}}$ by ${(- 6)}^{\frac{3}{2}}$ and simplify.

Divide each term in $x {(- 6)}^{\frac{3}{2}} = {(- 5)}^{\frac{3}{2}}$ by ${(- 6)}^{\frac{3}{2}}$ .

$\frac{x {(- 6)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}} = \frac{{(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x {(- 6)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}} = \frac{{(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

Divide $x$ by $1$ .

$x = \frac{{(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

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

$x {(- 6)}^{\frac{3}{2}} = - {(- 5)}^{\frac{3}{2}}$

Divide each term in $x {(- 6)}^{\frac{3}{2}} = - {(- 5)}^{\frac{3}{2}}$ by ${(- 6)}^{\frac{3}{2}}$ and simplify.

Divide each term in $x {(- 6)}^{\frac{3}{2}} = - {(- 5)}^{\frac{3}{2}}$ by ${(- 6)}^{\frac{3}{2}}$ .

$\frac{x {(- 6)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}} = \frac{- {(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x {(- 6)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}} = \frac{- {(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

Divide $x$ by $1$ .

$x = \frac{- {(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{{(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

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

$x = \frac{{(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}, - \frac{{(- 5)}^{\frac{3}{2}}}{{(- 6)}^{\frac{3}{2}}}$

Step 7

Exclude the solutions that do not make $- 8 = - 3 - 6 \sqrt[3]{x^{2}}$ true.

$No solution$

Evaluate -8=-3-6 cube root of x^2

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