Solve for x fifth root of 9x+4 = fifth root of 5x-12

$\sqrt[5]{9 x + 4} = \sqrt[5]{5 x - 12}$

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{9 x + 4}}^{5} = {\sqrt[5]{5 x - 12}}^{5}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{9 x + 4}$ as ${(9 x + 4)}^{\frac{1}{5}}$ .

${({(9 x + 4)}^{\frac{1}{5}})}^{5} = {\sqrt[5]{5 x - 12}}^{5}$

Simplify the left side.

Simplify ${({(9 x + 4)}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(9 x + 4)}^{\frac{1}{5}})}^{5}$ .

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

${(9 x + 4)}^{\frac{1}{5} \cdot 5} = {\sqrt[5]{5 x - 12}}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(9 x + 4)}^{\frac{1}{5} \cdot 5} = {\sqrt[5]{5 x - 12}}^{5}$

Rewrite the expression.

${(9 x + 4)}^{1} = {\sqrt[5]{5 x - 12}}^{5}$

Simplify.

$9 x + 4 = {\sqrt[5]{5 x - 12}}^{5}$

Simplify the right side.

Rewrite ${\sqrt[5]{5 x - 12}}^{5}$ as $5 x - 12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{5 x - 12}$ as ${(5 x - 12)}^{\frac{1}{5}}$ .

$9 x + 4 = {({(5 x - 12)}^{\frac{1}{5}})}^{5}$

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

$9 x + 4 = {(5 x - 12)}^{\frac{1}{5} \cdot 5}$

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

$9 x + 4 = {(5 x - 12)}^{\frac{5}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$9 x + 4 = {(5 x - 12)}^{\frac{5}{5}}$

Rewrite the expression.

$9 x + 4 = {(5 x - 12)}^{1}$

Simplify.

$9 x + 4 = 5 x - 12$

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $5 x$ from both sides of the equation.

$9 x + 4 - 5 x = - 12$

Subtract $5 x$ from $9 x$ .

$4 x + 4 = - 12$

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

Subtract $4$ from both sides of the equation.

$4 x = - 12 - 4$

Subtract $4$ from $- 12$ .

$4 x = - 16$

Divide each term in $4 x = - 16$ by $4$ and simplify.

Divide each term in $4 x = - 16$ by $4$ .

$\frac{4 x}{4} = \frac{- 16}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 16}{4}$

Divide $x$ by $1$ .

$x = \frac{- 16}{4}$

Simplify the right side.

Divide $- 16$ by $4$ .

$x = - 4$

Solve for x fifth root of 9x+4 = fifth root of 5x-12

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