Algebra Examples

$\frac{\sqrt{x} + 5}{\sqrt{x} - 5}$

Step 1

Multiply to rationalize the numerator.

$\frac{(\sqrt{x} + 5) (\sqrt{x} - 5)}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$\frac{{\sqrt{x}}^{2} + \sqrt{x} \cdot - 5 + \sqrt{x} \cdot 5 - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2.2

Simplify.

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Step 2.2.1

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

$\frac{{(x^{\frac{1}{2}})}^{2} - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2.2.2

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

$\frac{x^{\frac{1}{2} \cdot 2} - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2.2.3

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

$\frac{x^{\frac{2}{2}} - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\frac{x^{\frac{2}{2}} - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2.2.4.2

Rewrite the expression.

$\frac{x^{1} - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 2.2.5

Simplify.

$\frac{x - 25}{(\sqrt{x} - 5) (\sqrt{x} - 5)}$

Step 3

Expand $(\sqrt{x} - 5) (\sqrt{x} - 5)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{x - 25}{\sqrt{x} (\sqrt{x} - 5) - 5 (\sqrt{x} - 5)}$

Step 3.2

Apply the distributive property.

$\frac{x - 25}{\sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 5 - 5 (\sqrt{x} - 5)}$

Step 3.3

Apply the distributive property.

$\frac{x - 25}{\sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

$\frac{x - 25}{{\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.1.2

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

$\frac{x - 25}{{\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.1.3

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

$\frac{x - 25}{{\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.1.4

Add $1$ and $1$ .

$\frac{x - 25}{{\sqrt{x}}^{2} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.2

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

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Step 4.1.2.1

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

$\frac{x - 25}{{(x^{\frac{1}{2}})}^{2} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.2.2

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

$\frac{x - 25}{x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.2.3

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

$\frac{x - 25}{x^{\frac{2}{2}} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

$\frac{x - 25}{x^{\frac{2}{2}} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.2.4.2

Rewrite the expression.

$\frac{x - 25}{x^{1} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.2.5

Simplify.

$\frac{x - 25}{x + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.3

Move $- 5$ to the left of $\sqrt{x}$ .

$\frac{x - 25}{x - 5 \cdot \sqrt{x} - 5 \sqrt{x} - 5 \cdot - 5}$

Step 4.1.4

Multiply $- 5$ by $- 5$ .

$\frac{x - 25}{x - 5 \sqrt{x} - 5 \sqrt{x} + 25}$

Step 4.2

Subtract $5 \sqrt{x}$ from $- 5 \sqrt{x}$ .

$\frac{x - 25}{x - 10 \sqrt{x} + 25}$