Algebra Examples

$\frac{x^{2} - 16}{\sqrt{x} - 2}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 4^{2}}{\sqrt{x} - 2}$

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

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

Step 3.3.5

Simplify.

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

Step 3.4

Cancel the common factor of $x - 4$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

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

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

Step 4

Expand $(x + 4) (\sqrt{x} + 2)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify each term.

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Step 5.1

Move $2$ to the left of $x$ .

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

Step 5.2

Multiply $4$ by $2$ .

$x \sqrt{x} + 2 x + 4 \sqrt{x} + 8$