Finite Math Examples

$y = \sqrt[3]{x} - 6$

Step 1

Rewrite the equation as $\sqrt[3]{x} - 6 = y$ .

$\sqrt[3]{x} - 6 = y$

Step 2

Add $6$ to both sides of the equation.

$\sqrt[3]{x} = y + 6$

Step 3

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

${\sqrt[3]{x}}^{3} = {(y + 6)}^{3}$

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(x^{\frac{1}{3}})}^{3} = {(y + 6)}^{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(y + 6)}^{3}$

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.

$x^{\frac{1}{3} \cdot 3} = {(y + 6)}^{3}$

Step 4.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(y + 6)}^{3}$

Step 4.2.1.2

Simplify.

$x = {(y + 6)}^{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(y + 6)}^{3}$ .

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Step 4.3.1.1

Use the Binomial Theorem.

$x = y^{3} + 3 y^{2} \cdot 6 + 3 y \cdot 6^{2} + 6^{3}$

Step 4.3.1.2

Simplify each term.

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Step 4.3.1.2.1

Multiply $6$ by $3$ .

$x = y^{3} + 18 y^{2} + 3 y \cdot 6^{2} + 6^{3}$

Step 4.3.1.2.2

Raise $6$ to the power of $2$ .

$x = y^{3} + 18 y^{2} + 3 y \cdot 36 + 6^{3}$

Step 4.3.1.2.3

Multiply $36$ by $3$ .

$x = y^{3} + 18 y^{2} + 108 y + 6^{3}$

Step 4.3.1.2.4

Raise $6$ to the power of $3$ .

$x = y^{3} + 18 y^{2} + 108 y + 216$