Solve for a cube root of 3a+9 = cube root of 6a+15

$\sqrt[3]{3 a + 9} = \sqrt[3]{6 a + 15}$

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

${\sqrt[3]{3 a + 9}}^{3} = {\sqrt[3]{6 a + 15}}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 a + 9}$ as ${(3 a + 9)}^{\frac{1}{3}}$ .

${({(3 a + 9)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{6 a + 15}}^{3}$

Simplify the left side.

Simplify ${({(3 a + 9)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(3 a + 9)}^{\frac{1}{3}})}^{3}$ .

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

${(3 a + 9)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{6 a + 15}}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(3 a + 9)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{6 a + 15}}^{3}$

Rewrite the expression.

${(3 a + 9)}^{1} = {\sqrt[3]{6 a + 15}}^{3}$

Simplify.

$3 a + 9 = {\sqrt[3]{6 a + 15}}^{3}$

Simplify the right side.

Simplify ${\sqrt[3]{6 a + 15}}^{3}$ .

Factor $3$ out of $6 a + 15$ .

Factor $3$ out of $6 a$ .

$3 a + 9 = {\sqrt[3]{3 (2 a) + 15}}^{3}$

Factor $3$ out of $15$ .

$3 a + 9 = {\sqrt[3]{3 (2 a) + 3 (5)}}^{3}$

Factor $3$ out of $3 (2 a) + 3 (5)$ .

$3 a + 9 = {\sqrt[3]{3 (2 a + 5)}}^{3}$

Rewrite ${\sqrt[3]{3 (2 a + 5)}}^{3}$ as $3 (2 a + 5)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 (2 a + 5)}$ as ${(3 (2 a + 5))}^{\frac{1}{3}}$ .

$3 a + 9 = {({(3 (2 a + 5))}^{\frac{1}{3}})}^{3}$

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

$3 a + 9 = {(3 (2 a + 5))}^{\frac{1}{3} \cdot 3}$

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

$3 a + 9 = {(3 (2 a + 5))}^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 a + 9 = {(3 (2 a + 5))}^{\frac{3}{3}}$

Rewrite the expression.

$3 a + 9 = {(3 (2 a + 5))}^{1}$

Simplify.

$3 a + 9 = 3 (2 a + 5)$

Apply the distributive property.

$3 a + 9 = 3 (2 a) + 3 \cdot 5$

Multiply.

Multiply $2$ by $3$ .

$3 a + 9 = 6 a + 3 \cdot 5$

Multiply $3$ by $5$ .

$3 a + 9 = 6 a + 15$

Solve for $a$ .

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

Subtract $6 a$ from both sides of the equation.

$3 a + 9 - 6 a = 15$

Subtract $6 a$ from $3 a$ .

$- 3 a + 9 = 15$

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

Subtract $9$ from both sides of the equation.

$- 3 a = 15 - 9$

Subtract $9$ from $15$ .

$- 3 a = 6$

Divide each term in $- 3 a = 6$ by $- 3$ and simplify.

Divide each term in $- 3 a = 6$ by $- 3$ .

$\frac{- 3 a}{- 3} = \frac{6}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 a}{- 3} = \frac{6}{- 3}$

Divide $a$ by $1$ .

$a = \frac{6}{- 3}$

Simplify the right side.

Divide $6$ by $- 3$ .

$a = - 2$

Solve for a cube root of 3a+9 = cube root of 6a+15

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