Pre-Algebra Examples

$\frac{1}{\sqrt{x} - 1} - \frac{1}{\sqrt{x} + 1} + \frac{1}{x - 1}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{\sqrt{x} - 1}$ by $\frac{\sqrt{x} + 1}{\sqrt{x} + 1}$ .

$\frac{1}{\sqrt{x} - 1} \cdot \frac{\sqrt{x} + 1}{\sqrt{x} + 1} - \frac{1}{\sqrt{x} + 1} + \frac{1}{x - 1}$

Step 1.2

Multiply $\frac{1}{\sqrt{x} - 1}$ by $\frac{\sqrt{x} + 1}{\sqrt{x} + 1}$ .

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

Step 1.3

Expand the denominator using the FOIL method.

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Cancel the common factor.

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

Step 1.4.4.2

Rewrite the expression.

$\frac{\sqrt{x} + 1}{x^{1} - 1} - \frac{1}{\sqrt{x} + 1} + \frac{1}{x - 1}$

Step 1.4.5

Simplify.

$\frac{\sqrt{x} + 1}{x - 1} - \frac{1}{\sqrt{x} + 1} + \frac{1}{x - 1}$

Step 1.5

Multiply $\frac{1}{\sqrt{x} + 1}$ by $\frac{\sqrt{x} - 1}{\sqrt{x} - 1}$ .

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

Step 1.6

Multiply $\frac{1}{\sqrt{x} + 1}$ by $\frac{\sqrt{x} - 1}{\sqrt{x} - 1}$ .

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

Step 1.7

Expand the denominator using the FOIL method.

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

Step 1.8

Simplify.

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

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

Step 1.8.4.2

Rewrite the expression.

$\frac{\sqrt{x} + 1}{x - 1} - \frac{\sqrt{x} - 1}{x^{1} - 1} + \frac{1}{x - 1}$

Step 1.8.5

Simplify.

$\frac{\sqrt{x} + 1}{x - 1} - \frac{\sqrt{x} - 1}{x - 1} + \frac{1}{x - 1}$

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{x} + 1 - \sqrt{x} - - 1 + 1}{x - 1}$

Step 3.2

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{x} + 1 - \sqrt{x} + 1 + 1}{x - 1}$

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $\sqrt{x} + 1 - \sqrt{x} + 1 + 1$ .

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Step 4.1.1

Subtract $\sqrt{x}$ from $\sqrt{x}$ .

$\frac{0 + 1 + 1 + 1}{x - 1}$

Step 4.1.2

Add $0$ and $1$ .

$\frac{1 + 1 + 1}{x - 1}$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $1$ and $1$ .

$\frac{2 + 1}{x - 1}$

Step 4.2.2

Add $2$ and $1$ .

$\frac{3}{x - 1}$