Algebra Examples

$\frac{\sqrt{3} + 5}{\sqrt{3} - 5}$

Step 1

Multiply $\frac{\sqrt{3} + 5}{\sqrt{3} - 5}$ by $\frac{\sqrt{3} + 5}{\sqrt{3} + 5}$ .

$\frac{\sqrt{3} + 5}{\sqrt{3} - 5} \cdot \frac{\sqrt{3} + 5}{\sqrt{3} + 5}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{3} + 5}{\sqrt{3} - 5}$ by $\frac{\sqrt{3} + 5}{\sqrt{3} + 5}$ .

$\frac{(\sqrt{3} + 5) (\sqrt{3} + 5)}{(\sqrt{3} - 5) (\sqrt{3} + 5)}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} + 5) (\sqrt{3} + 5)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot 5 - 5 \sqrt{3} - 25}$

Step 2.3

Simplify.

$\frac{(\sqrt{3} + 5) (\sqrt{3} + 5)}{- 22}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{3} + 5$ to the power of $1$ .

$\frac{{(\sqrt{3} + 5)}^{1} (\sqrt{3} + 5)}{- 22}$

Step 3.2

Raise $\sqrt{3} + 5$ to the power of $1$ .

$\frac{{(\sqrt{3} + 5)}^{1} {(\sqrt{3} + 5)}^{1}}{- 22}$

Step 3.3

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

$\frac{{(\sqrt{3} + 5)}^{1 + 1}}{- 22}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{3} + 5)}^{2}}{- 22}$

Step 4

Move the negative in front of the fraction.

$- \frac{{(\sqrt{3} + 5)}^{2}}{22}$

Step 5

Simplify ${(\sqrt{3} + 5)}^{2}$ .

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Step 5.1

Rewrite ${(\sqrt{3} + 5)}^{2}$ as $(\sqrt{3} + 5) (\sqrt{3} + 5)$ .

$- \frac{(\sqrt{3} + 5) (\sqrt{3} + 5)}{22}$

Step 5.2

Expand $(\sqrt{3} + 5) (\sqrt{3} + 5)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$- \frac{\sqrt{3} (\sqrt{3} + 5) + 5 (\sqrt{3} + 5)}{22}$

Step 5.2.2

Apply the distributive property.

$- \frac{\sqrt{3} \sqrt{3} + \sqrt{3} \cdot 5 + 5 (\sqrt{3} + 5)}{22}$

Step 5.2.3

Apply the distributive property.

$- \frac{\sqrt{3} \sqrt{3} + \sqrt{3} \cdot 5 + 5 \sqrt{3} + 5 \cdot 5}{22}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Combine using the product rule for radicals.

$- \frac{\sqrt{3 \cdot 3} + \sqrt{3} \cdot 5 + 5 \sqrt{3} + 5 \cdot 5}{22}$

Step 5.3.1.2

Multiply $3$ by $3$ .

$- \frac{\sqrt{9} + \sqrt{3} \cdot 5 + 5 \sqrt{3} + 5 \cdot 5}{22}$

Step 5.3.1.3

Rewrite $9$ as $3^{2}$ .

$- \frac{\sqrt{3^{2}} + \sqrt{3} \cdot 5 + 5 \sqrt{3} + 5 \cdot 5}{22}$

Step 5.3.1.4

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{3 + \sqrt{3} \cdot 5 + 5 \sqrt{3} + 5 \cdot 5}{22}$

Step 5.3.1.5

Move $5$ to the left of $\sqrt{3}$ .

$- \frac{3 + 5 \cdot \sqrt{3} + 5 \sqrt{3} + 5 \cdot 5}{22}$

Step 5.3.1.6

Multiply $5$ by $5$ .

$- \frac{3 + 5 \sqrt{3} + 5 \sqrt{3} + 25}{22}$

Step 5.3.2

Add $3$ and $25$ .

$- \frac{28 + 5 \sqrt{3} + 5 \sqrt{3}}{22}$

Step 5.3.3

Add $5 \sqrt{3}$ and $5 \sqrt{3}$ .

$- \frac{28 + 10 \sqrt{3}}{22}$

Step 6

Cancel the common factor of $28 + 10 \sqrt{3}$ and $22$ .

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Step 6.1

Factor $2$ out of $28$ .

$- \frac{2 (14) + 10 \sqrt{3}}{22}$

Step 6.2

Factor $2$ out of $10 \sqrt{3}$ .

$- \frac{2 (14) + 2 (5 \sqrt{3})}{22}$

Step 6.3

Factor $2$ out of $2 (14) + 2 (5 \sqrt{3})$ .

$- \frac{2 (14 + 5 \sqrt{3})}{22}$

Step 6.4

Cancel the common factors.

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Step 6.4.1

Factor $2$ out of $22$ .

$- \frac{2 (14 + 5 \sqrt{3})}{2 \cdot 11}$

Step 6.4.2

Cancel the common factor.

$- \frac{2 (14 + 5 \sqrt{3})}{2 \cdot 11}$

Step 6.4.3

Rewrite the expression.

$- \frac{14 + 5 \sqrt{3}}{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{14 + 5 \sqrt{3}}{11}$

Decimal Form:

$- 2.06002309 \dots$