Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} (\sqrt{5} + \sqrt{3})}{{\sqrt{5}}^{2} + \sqrt{15} - \sqrt{15} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Combine using the product rule for radicals.

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

Step 2.6

Combine using the product rule for radicals.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $5$ by $5$ .

$\frac{\sqrt{25} + \sqrt{5 \cdot 3}}{2}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $5$ by $3$ .

$\frac{5 + \sqrt{15}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{5 + \sqrt{15}}{2}$

Decimal Form:

$4.43649167 \dots$