Algebra Examples

$f (x) = \sqrt{x}$ , $g (x) = \sqrt{x} + 2$

Step 1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{\sqrt{x}}{\sqrt{x} + 2}$

Step 2

Simplify.

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Step 2.1

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

$\frac{\sqrt{x}}{\sqrt{x} + 2} \cdot \frac{\sqrt{x} - 2}{\sqrt{x} - 2}$

Step 2.2

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

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

Step 2.3

Expand the denominator using the FOIL method.

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Simplify.

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

Step 2.5

Apply the distributive property.

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

Step 2.6

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 2.6.1

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 2.6.2

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 2.6.3

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

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

Step 2.6.4

Add $1$ and $1$ .

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

Step 2.7

Simplify terms.

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Step 2.7.1

Move $- 2$ to the left of $\sqrt{x}$ .

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

Step 2.7.2

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

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Step 2.7.2.1

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

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

Step 2.7.2.2

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

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

Step 2.7.2.3

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

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

Step 2.7.2.4

Cancel the common factor of $2$ .

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Step 2.7.2.4.1

Cancel the common factor.

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

Step 2.7.2.4.2

Rewrite the expression.

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

Step 2.7.2.5

Simplify.

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

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

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

$\frac{x - 2 x^{\frac{1}{2}}}{x - 4}$

Step 2.8.2

Factor $x^{\frac{1}{2}}$ out of $x - 2 x^{\frac{1}{2}}$ .

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Step 2.8.2.1

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

$\frac{x^{1} - 2 x^{\frac{1}{2}}}{x - 4}$

Step 2.8.2.2

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

$\frac{x^{\frac{1}{2}} x^{\frac{1}{2}} - 2 x^{\frac{1}{2}}}{x - 4}$

Step 2.8.2.3

Factor $x^{\frac{1}{2}}$ out of $- 2 x^{\frac{1}{2}}$ .

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

Step 2.8.2.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot - 2$ .

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