Basic Math Examples

$\frac{4 + \sqrt{5}}{3 - \sqrt{5}}$

Step 1

Multiply $\frac{4 + \sqrt{5}}{3 - \sqrt{5}}$ by $\frac{3 + \sqrt{5}}{3 + \sqrt{5}}$ .

$\frac{4 + \sqrt{5}}{3 - \sqrt{5}} \cdot \frac{3 + \sqrt{5}}{3 + \sqrt{5}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{4 + \sqrt{5}}{3 - \sqrt{5}}$ by $\frac{3 + \sqrt{5}}{3 + \sqrt{5}}$ .

$\frac{(4 + \sqrt{5}) (3 + \sqrt{5})}{(3 - \sqrt{5}) (3 + \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(4 + \sqrt{5}) (3 + \sqrt{5})}{9 + 3 \sqrt{5} - 3 \sqrt{5} - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(4 + \sqrt{5}) (3 + \sqrt{5})}{4}$

Step 3

Expand $(4 + \sqrt{5}) (3 + \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{4 (3 + \sqrt{5}) + \sqrt{5} (3 + \sqrt{5})}{4}$

Step 3.2

Apply the distributive property.

$\frac{4 \cdot 3 + 4 \sqrt{5} + \sqrt{5} (3 + \sqrt{5})}{4}$

Step 3.3

Apply the distributive property.

$\frac{4 \cdot 3 + 4 \sqrt{5} + \sqrt{5} \cdot 3 + \sqrt{5} \sqrt{5}}{4}$

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 $4$ by $3$ .

$\frac{12 + 4 \sqrt{5} + \sqrt{5} \cdot 3 + \sqrt{5} \sqrt{5}}{4}$

Step 4.1.2

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

$\frac{12 + 4 \sqrt{5} + 3 \cdot \sqrt{5} + \sqrt{5} \sqrt{5}}{4}$

Step 4.1.3

Combine using the product rule for radicals.

$\frac{12 + 4 \sqrt{5} + 3 \sqrt{5} + \sqrt{5 \cdot 5}}{4}$

Step 4.1.4

Multiply $5$ by $5$ .

$\frac{12 + 4 \sqrt{5} + 3 \sqrt{5} + \sqrt{25}}{4}$

Step 4.1.5

Rewrite $25$ as $5^{2}$ .

$\frac{12 + 4 \sqrt{5} + 3 \sqrt{5} + \sqrt{5^{2}}}{4}$

Step 4.1.6

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

$\frac{12 + 4 \sqrt{5} + 3 \sqrt{5} + 5}{4}$

Step 4.2

Add $12$ and $5$ .

$\frac{17 + 4 \sqrt{5} + 3 \sqrt{5}}{4}$

Step 4.3

Add $4 \sqrt{5}$ and $3 \sqrt{5}$ .

$\frac{17 + 7 \sqrt{5}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{17 + 7 \sqrt{5}}{4}$

Decimal Form:

$8.16311896 \dots$