Solve the Rational Equation for x 1/5 cube root of 3x+10=8

$\frac{1}{5} \sqrt[3]{3 x} + 10 = 8$

Move all terms not containing $\frac{1}{5} \cdot \sqrt[3]{3 x}$ to the right side of the equation.

Subtract $10$ from both sides of the equation.

$\frac{1}{5} \cdot \sqrt[3]{3 x} = 8 - 10$

Subtract $10$ from $8$ .

$\frac{1}{5} \cdot \sqrt[3]{3 x} = - 2$

To remove the radical on the left side of the equation, cube both sides of the equation.

${(\frac{1}{5} \cdot \sqrt[3]{3 x})}^{3} = {(- 2)}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 x}$ as ${(3 x)}^{\frac{1}{3}}$ .

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

Simplify the left side.

Simplify ${(\frac{1}{5} \cdot {(3 x)}^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $3 x$ .

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

Multiply $\frac{1}{5} (3^{\frac{1}{3}} x^{\frac{1}{3}})$ .

Combine $3^{\frac{1}{3}}$ and $\frac{1}{5}$ .

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

Combine $\frac{3^{\frac{1}{3}}}{5}$ and $x^{\frac{1}{3}}$ .

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{3^{\frac{1}{3}} x^{\frac{1}{3}}}{5}$ .

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

Apply the product rule to $3^{\frac{1}{3}} x^{\frac{1}{3}}$ .

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

Simplify the numerator.

Multiply the exponents in ${(3^{\frac{1}{3}})}^{3}$ .

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Raise $5$ to the power of $3$ .

$\frac{3 x}{125} = {(- 2)}^{3}$

Simplify the right side.

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

$\frac{3 x}{125} = - 8$

Solve for $x$ .

Multiply both sides of the equation by $\frac{125}{3}$ .

$\frac{125}{3} \cdot \frac{3 x}{125} = \frac{125}{3} \cdot - 8$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{125}{3} \cdot \frac{3 x}{125}$ .

Cancel the common factor of $125$ .

Cancel the common factor.

$\frac{125}{3} \cdot \frac{3 x}{125} = \frac{125}{3} \cdot - 8$

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{125}{3} \cdot - 8$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{125}{3} \cdot - 8$

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{125}{3} \cdot - 8$

Rewrite the expression.

$x = \frac{125}{3} \cdot - 8$

Simplify the right side.

Simplify $\frac{125}{3} \cdot - 8$ .

Multiply $\frac{125}{3} \cdot - 8$ .

Combine $\frac{125}{3}$ and $- 8$ .

$x = \frac{125 \cdot - 8}{3}$

Multiply $125$ by $- 8$ .

$x = \frac{- 1000}{3}$

Move the negative in front of the fraction.

$x = - \frac{1000}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1000}{3}$

Decimal Form:

$x = - 333.\overline{3}$

Mixed Number Form:

$x = - 333 \frac{1}{3}$

Solve the Rational Equation for x 1/5 cube root of 3x+10=8

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