Algebra Examples

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

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

Step 1

Multiply to rationalize the numerator.

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

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

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

Step 2.2

Simplify.

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Step 2.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ .

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

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

Step 2.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 2.2.4

Cancel the common factor of

2

$2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Simplify.

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

Step 3

Expand

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify each term.

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Step 4.1

Multiply

$x$ by

$1$ .

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

Step 4.2

Rewrite

$- 1 \sqrt{x}$ as

$- \sqrt{x}$ .

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

Step 4.3

Multiply

$- 1$ by

$1$ .

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