Algebra Examples

$\sqrt[3]{4 + 6 t} - \sqrt[3]{1 - 8 t} = 0$

Step 1

Factor $2$ out of $4 + 6 t$ .

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Step 1.1

Factor $2$ out of $4$ .

$\sqrt[3]{2 (2) + 6 t} - \sqrt[3]{1 - 8 t} = 0$

Step 1.2

Factor $2$ out of $6 t$ .

$\sqrt[3]{2 (2) + 2 (3 t)} - \sqrt[3]{1 - 8 t} = 0$

Step 1.3

Factor $2$ out of $2 (2) + 2 (3 t)$ .

$\sqrt[3]{2 (2 + 3 t)} - \sqrt[3]{1 - 8 t} = 0$

Step 2

Add $\sqrt[3]{1 - 8 t}$ to both sides of the equation.

$\sqrt[3]{2 (2 + 3 t)} = \sqrt[3]{1 - 8 t}$

Step 3

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

${\sqrt[3]{2 (2 + 3 t)}}^{3} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${({(2 (2 + 3 t))}^{\frac{1}{3}})}^{3} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(2 (2 + 3 t))}^{\frac{1}{3}})}^{3}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(2 (2 + 3 t))}^{\frac{1}{3}})}^{3}$ .

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Step 4.2.1.1.1

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

${(2 (2 + 3 t))}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(2 (2 + 3 t))}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(2 (2 + 3 t))}^{1} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2.1.2

Apply the distributive property.

${(2 \cdot 2 + 2 (3 t))}^{1} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2.1.3

Multiply.

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Step 4.2.1.3.1

Multiply $2$ by $2$ .

${(4 + 2 (3 t))}^{1} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2.1.3.2

Multiply $3$ by $2$ .

${(4 + 6 t)}^{1} = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.2.1.3.3

Simplify.

$4 + 6 t = {\sqrt[3]{1 - 8 t}}^{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Rewrite ${\sqrt[3]{1 - 8 t}}^{3}$ as $1 - 8 t$ .

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Step 4.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{1 - 8 t}$ as ${(1 - 8 t)}^{\frac{1}{3}}$ .

$4 + 6 t = {({(1 - 8 t)}^{\frac{1}{3}})}^{3}$

Step 4.3.1.2

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

$4 + 6 t = {(1 - 8 t)}^{\frac{1}{3} \cdot 3}$

Step 4.3.1.3

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

$4 + 6 t = {(1 - 8 t)}^{\frac{3}{3}}$

Step 4.3.1.4

Cancel the common factor of $3$ .

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Step 4.3.1.4.1

Cancel the common factor.

$4 + 6 t = {(1 - 8 t)}^{\frac{3}{3}}$

Step 4.3.1.4.2

Rewrite the expression.

$4 + 6 t = {(1 - 8 t)}^{1}$

Step 4.3.1.5

Simplify.

$4 + 6 t = 1 - 8 t$

Step 5

Solve for $t$ .

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Step 5.1

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

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Step 5.1.1

Add $8 t$ to both sides of the equation.

$4 + 6 t + 8 t = 1$

Step 5.1.2

Add $6 t$ and $8 t$ .

$4 + 14 t = 1$

Step 5.2

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

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Step 5.2.1

Subtract $4$ from both sides of the equation.

$14 t = 1 - 4$

Step 5.2.2

Subtract $4$ from $1$ .

$14 t = - 3$

Step 5.3

Divide each term in $14 t = - 3$ by $14$ and simplify.

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Step 5.3.1

Divide each term in $14 t = - 3$ by $14$ .

$\frac{14 t}{14} = \frac{- 3}{14}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $14$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{14 t}{14} = \frac{- 3}{14}$

Step 5.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 3}{14}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

$t = - \frac{3}{14}$