Linear Algebra Examples

$\frac{3 + \sqrt{5}}{2 - \sqrt{5}}$

Step 1

Multiply $\frac{3 + \sqrt{5}}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{3 + \sqrt{5}}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(3 + \sqrt{5}) (2 + \sqrt{5})}{- 1}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of $\frac{(3 + \sqrt{5}) (2 + \sqrt{5})}{- 1}$ .

$- 1 \cdot ((3 + \sqrt{5}) (2 + \sqrt{5}))$

Step 2.4.2

Rewrite $- 1 \cdot ((3 + \sqrt{5}) (2 + \sqrt{5}))$ as $- ((3 + \sqrt{5}) (2 + \sqrt{5}))$ .

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

Step 3

Expand $(3 + \sqrt{5}) (2 + \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $3$ by $2$ .

$- (6 + 3 \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \sqrt{5})$

Step 4.1.2

Move $2$ to the left of $\sqrt{5}$ .

$- (6 + 3 \sqrt{5} + 2 \cdot \sqrt{5} + \sqrt{5} \sqrt{5})$

Step 4.1.3

Combine using the product rule for radicals.

$- (6 + 3 \sqrt{5} + 2 \sqrt{5} + \sqrt{5 \cdot 5})$

Step 4.1.4

Multiply $5$ by $5$ .

$- (6 + 3 \sqrt{5} + 2 \sqrt{5} + \sqrt{25})$

Step 4.1.5

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

$- (6 + 3 \sqrt{5} + 2 \sqrt{5} + \sqrt{5^{2}})$

Step 4.1.6

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

$- (6 + 3 \sqrt{5} + 2 \sqrt{5} + 5)$

Step 4.2

Add $6$ and $5$ .

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

Step 4.3

Add $3 \sqrt{5}$ and $2 \sqrt{5}$ .

$- (11 + 5 \sqrt{5})$

Step 5

Apply the distributive property.

$- 1 \cdot 11 - (5 \sqrt{5})$

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $11$ .

$- 11 - (5 \sqrt{5})$

Step 6.2

Multiply $5$ by $- 1$ .

$- 11 - 5 \sqrt{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 11 - 5 \sqrt{5}$

Decimal Form:

$- 22.18033988 \dots$