Precalculus Examples

$\frac{x^{6} - 8}{x - \sqrt{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{{(x^{2})}^{3} - 8}{x - \sqrt{2}}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$\frac{{(x^{2})}^{3} - 2^{3}}{x - \sqrt{2}}$

Step 1.3

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

$\frac{(x^{2} - 2) ({(x^{2})}^{2} + x^{2} \cdot 2 + 2^{2})}{x - \sqrt{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.4.1.1

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

$\frac{(x^{2} - 2) (x^{2 \cdot 2} + x^{2} \cdot 2 + 2^{2})}{x - \sqrt{2}}$

Step 1.4.1.2

Multiply $2$ by $2$ .

$\frac{(x^{2} - 2) (x^{4} + x^{2} \cdot 2 + 2^{2})}{x - \sqrt{2}}$

Step 1.4.2

Move $2$ to the left of $x^{2}$ .

$\frac{(x^{2} - 2) (x^{4} + 2 \cdot x^{2} + 2^{2})}{x - \sqrt{2}}$

Step 1.4.3

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

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4)}{x - \sqrt{2}}$

Step 2

Multiply $\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4)}{x - \sqrt{2}}$ by $\frac{x + \sqrt{2}}{x + \sqrt{2}}$ .

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4)}{x - \sqrt{2}} \cdot \frac{x + \sqrt{2}}{x + \sqrt{2}}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4)}{x - \sqrt{2}}$ by $\frac{x + \sqrt{2}}{x + \sqrt{2}}$ .

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4) (x + \sqrt{2})}{(x - \sqrt{2}) (x + \sqrt{2})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4) (x + \sqrt{2})}{x^{2} + x \sqrt{2} - \sqrt{2} x - {\sqrt{2}}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4) (x + \sqrt{2})}{x^{2} - 2^{\frac{2}{2}}}$

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Cancel the common factor.

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4) (x + \sqrt{2})}{x^{2} - 2^{\frac{2}{2}}}$

Step 3.3.1.4.2

Rewrite the expression.

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

Step 3.3.1.5

Evaluate the exponent.

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

Step 3.3.2

Multiply $- 1$ by $2$ .

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4) (x + \sqrt{2})}{x^{2} - 2}$

Step 3.4

Cancel the common factor of $x^{2} - 2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(x^{2} - 2) (x^{4} + 2 x^{2} + 4) (x + \sqrt{2})}{x^{2} - 2}$

Step 3.4.2

Divide $(x^{4} + 2 x^{2} + 4) (x + \sqrt{2})$ by $1$ .

$(x^{4} + 2 x^{2} + 4) (x + \sqrt{2})$

Step 4

Expand $(x^{4} + 2 x^{2} + 4) (x + \sqrt{2})$ by multiplying each term in the first expression by each term in the second expression.

$x^{4} x + x^{4} \sqrt{2} + 2 x^{2} x + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5

Simplify each term.

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Step 5.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 5.1.1

Multiply $x^{4}$ by $x$ .

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Step 5.1.1.1

Raise $x$ to the power of $1$ .

$x^{4} x^{1} + x^{4} \sqrt{2} + 2 x^{2} x + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4 + 1} + x^{4} \sqrt{2} + 2 x^{2} x + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5.1.2

Add $4$ and $1$ .

$x^{5} + x^{4} \sqrt{2} + 2 x^{2} x + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.2.1

Move $x$ .

$x^{5} + x^{4} \sqrt{2} + 2 (x \cdot x^{2}) + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5.2.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.2.1

Raise $x$ to the power of $1$ .

$x^{5} + x^{4} \sqrt{2} + 2 (x^{1} x^{2}) + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{5} + x^{4} \sqrt{2} + 2 x^{1 + 2} + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$

Step 5.2.3

Add $1$ and $2$ .

$x^{5} + x^{4} \sqrt{2} + 2 x^{3} + 2 x^{2} \sqrt{2} + 4 x + 4 \sqrt{2}$