Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

Expand $(\sqrt{x} + 1) (\sqrt{x} + 1)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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Step 8.1.1.1

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

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

Step 8.1.1.2

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

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

Step 8.1.1.3

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

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

Step 8.1.1.4

Add $1$ and $1$ .

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

Step 8.1.2

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

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Step 8.1.2.1

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

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

Step 8.1.2.2

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

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

Step 8.1.2.3

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

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

Step 8.1.2.4

Cancel the common factor of $2$ .

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Step 8.1.2.4.1

Cancel the common factor.

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

Step 8.1.2.4.2

Rewrite the expression.

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

Step 8.1.2.5

Simplify.

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

Step 8.1.3

Multiply $\sqrt{x}$ by $1$ .

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

Step 8.1.4

Multiply $\sqrt{x}$ by $1$ .

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

Step 8.1.5

Multiply $1$ by $1$ .

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

Step 8.2

Add $\sqrt{x}$ and $\sqrt{x}$ .

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

Step 9

Split the fraction $\frac{x + 2 \sqrt{x} + 1}{x - 1}$ into two fractions.

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

Step 10

Split the fraction $\frac{x + 2 \sqrt{x}}{x - 1}$ into two fractions.

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