Finite Math Examples

$\sqrt[3]{{(x + 2)}^{2}} = 16$

Step 1

Subtract $16$ from both sides of the equation.

$\sqrt[3]{{(x + 2)}^{2}} - 16 = 0$

Step 2

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

${(x + 2)}^{\frac{2}{3}} - 16 = 0$

Step 3

Rewrite ${(x + 2)}^{\frac{2}{3}}$ as ${({(x + 2)}^{\frac{1}{3}})}^{2}$ .

${({(x + 2)}^{\frac{1}{3}})}^{2} - 16 = 0$

Step 4

Rewrite $16$ as $4^{2}$ .

${({(x + 2)}^{\frac{1}{3}})}^{2} - 4^{2} = 0$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + 2)}^{\frac{1}{3}}$ and $b = 4$ .

$({(x + 2)}^{\frac{1}{3}} + 4) ({(x + 2)}^{\frac{1}{3}} - 4) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 2)}^{\frac{1}{3}} + 4 = 0$

${(x + 2)}^{\frac{1}{3}} - 4 = 0$

Step 7

Set ${(x + 2)}^{\frac{1}{3}} + 4$ equal to $0$ and solve for $x$ .

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Step 7.1

Set ${(x + 2)}^{\frac{1}{3}} + 4$ equal to $0$ .

${(x + 2)}^{\frac{1}{3}} + 4 = 0$

Step 7.2

Solve ${(x + 2)}^{\frac{1}{3}} + 4 = 0$ for $x$ .

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Step 7.2.1

Subtract $4$ from both sides of the equation.

${(x + 2)}^{\frac{1}{3}} = - 4$

Step 7.2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x + 2)}^{\frac{1}{3}})}^{3} = {(- 4)}^{3}$

Step 7.2.3

Simplify the exponent.

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Step 7.2.3.1

Simplify the left side.

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Step 7.2.3.1.1

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

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Step 7.2.3.1.1.1

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

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Step 7.2.3.1.1.1.1

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

${(x + 2)}^{\frac{1}{3} \cdot 3} = {(- 4)}^{3}$

Step 7.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.3.1.1.1.2.1

Cancel the common factor.

${(x + 2)}^{\frac{1}{3} \cdot 3} = {(- 4)}^{3}$

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

${(x + 2)}^{1} = {(- 4)}^{3}$

Step 7.2.3.1.1.2

Simplify.

$x + 2 = {(- 4)}^{3}$

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

Raise $- 4$ to the power of $3$ .

$x + 2 = - 64$

Step 7.2.4

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

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Step 7.2.4.1

Subtract $2$ from both sides of the equation.

$x = - 64 - 2$

Step 7.2.4.2

Subtract $2$ from $- 64$ .

$x = - 66$

Step 8

Set ${(x + 2)}^{\frac{1}{3}} - 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set ${(x + 2)}^{\frac{1}{3}} - 4$ equal to $0$ .

${(x + 2)}^{\frac{1}{3}} - 4 = 0$

Step 8.2

Solve ${(x + 2)}^{\frac{1}{3}} - 4 = 0$ for $x$ .

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Step 8.2.1

Add $4$ to both sides of the equation.

${(x + 2)}^{\frac{1}{3}} = 4$

Step 8.2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x + 2)}^{\frac{1}{3}})}^{3} = 4^{3}$

Step 8.2.3

Simplify the exponent.

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Step 8.2.3.1

Simplify the left side.

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Step 8.2.3.1.1

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

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Step 8.2.3.1.1.1

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

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Step 8.2.3.1.1.1.1

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

${(x + 2)}^{\frac{1}{3} \cdot 3} = 4^{3}$

Step 8.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 8.2.3.1.1.1.2.1

Cancel the common factor.

${(x + 2)}^{\frac{1}{3} \cdot 3} = 4^{3}$

Step 8.2.3.1.1.1.2.2

Rewrite the expression.

${(x + 2)}^{1} = 4^{3}$

Step 8.2.3.1.1.2

Simplify.

$x + 2 = 4^{3}$

Step 8.2.3.2

Simplify the right side.

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Step 8.2.3.2.1

Raise $4$ to the power of $3$ .

$x + 2 = 64$

Step 8.2.4

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

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Step 8.2.4.1

Subtract $2$ from both sides of the equation.

$x = 64 - 2$

Step 8.2.4.2

Subtract $2$ from $64$ .

$x = 62$

Step 9

The final solution is all the values that make $({(x + 2)}^{\frac{1}{3}} + 4) ({(x + 2)}^{\frac{1}{3}} - 4) = 0$ true.

$x = - 66, 62$