Simplify ( square root of 5+3)/(4- square root of 5)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} + 3) (4 + \sqrt{5})}{16 + 4 \sqrt{5} - 4 \sqrt{5} - {\sqrt{5}}^{2}}$

Simplify.

$\frac{(\sqrt{5} + 3) (4 + \sqrt{5})}{11}$

Step 3

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

Apply the distributive property.

$\frac{\sqrt{5} (4 + \sqrt{5}) + 3 (4 + \sqrt{5})}{11}$

Apply the distributive property.

$\frac{\sqrt{5} \cdot 4 + \sqrt{5} \sqrt{5} + 3 (4 + \sqrt{5})}{11}$

Apply the distributive property.

$\frac{\sqrt{5} \cdot 4 + \sqrt{5} \sqrt{5} + 3 \cdot 4 + 3 \sqrt{5}}{11}$

Step 4

Simplify and combine like terms.

Simplify each term.

Move $4$ to the left of $\sqrt{5}$ .

$\frac{4 \cdot \sqrt{5} + \sqrt{5} \sqrt{5} + 3 \cdot 4 + 3 \sqrt{5}}{11}$

Combine using the product rule for radicals.

$\frac{4 \sqrt{5} + \sqrt{5 \cdot 5} + 3 \cdot 4 + 3 \sqrt{5}}{11}$

Multiply $5$ by $5$ .

$\frac{4 \sqrt{5} + \sqrt{25} + 3 \cdot 4 + 3 \sqrt{5}}{11}$

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

$\frac{4 \sqrt{5} + \sqrt{5^{2}} + 3 \cdot 4 + 3 \sqrt{5}}{11}$

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

$\frac{4 \sqrt{5} + 5 + 3 \cdot 4 + 3 \sqrt{5}}{11}$

Multiply $3$ by $4$ .

$\frac{4 \sqrt{5} + 5 + 12 + 3 \sqrt{5}}{11}$

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

$\frac{5 + 12 + 7 \sqrt{5}}{11}$

Add $5$ and $12$ .

$\frac{17 + 7 \sqrt{5}}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{17 + 7 \sqrt{5}}{11}$

Decimal Form:

$2.96840689 \dots$