Algebra Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Factor $3$ out of $6 a$ .

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

Step 2.3.1.1.2

Factor $3$ out of $15$ .

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

Step 2.3.1.1.3

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

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

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

Step 2.3.1.2.4

Cancel the common factor of $3$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

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

Step 2.3.1.2.4.2

Rewrite the expression.

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

Step 2.3.1.2.5

Simplify.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Multiply.

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Step 2.3.1.4.1

Multiply $2$ by $3$ .

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

Step 2.3.1.4.2

Multiply $3$ by $5$ .

$3 a + 9 = 6 a + 15$

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

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

$3 a + 9 - 6 a = 15$

Step 3.1.2

Subtract $6 a$ from $3 a$ .

$- 3 a + 9 = 15$

Step 3.2

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

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Step 3.2.1

Subtract $9$ from both sides of the equation.

$- 3 a = 15 - 9$

Step 3.2.2

Subtract $9$ from $15$ .

$- 3 a = 6$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $a$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $6$ by $- 3$ .

$a = - 2$