Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

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

Step 6.1.2

Multiply $5$ by $3$ .

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

Step 6.1.3

Combine using the product rule for radicals.

$\frac{\sqrt{15} + \sqrt{5 \cdot 5} + \sqrt{3} \sqrt{3} + \sqrt{3} \sqrt{5}}{- 2}$

Step 6.1.4

Multiply $5$ by $5$ .

$\frac{\sqrt{15} + \sqrt{25} + \sqrt{3} \sqrt{3} + \sqrt{3} \sqrt{5}}{- 2}$

Step 6.1.5

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

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

Step 6.1.6

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

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

Step 6.1.7

Combine using the product rule for radicals.

$\frac{\sqrt{15} + 5 + \sqrt{3 \cdot 3} + \sqrt{3} \sqrt{5}}{- 2}$

Step 6.1.8

Multiply $3$ by $3$ .

$\frac{\sqrt{15} + 5 + \sqrt{9} + \sqrt{3} \sqrt{5}}{- 2}$

Step 6.1.9

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

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

Step 6.1.10

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

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

Step 6.1.11

Combine using the product rule for radicals.

$\frac{\sqrt{15} + 5 + 3 + \sqrt{3 \cdot 5}}{- 2}$

Step 6.1.12

Multiply $3$ by $5$ .

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

Step 6.2

Add $\sqrt{15}$ and $\sqrt{15}$ .

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

Step 6.3

Add $5$ and $3$ .

$\frac{8 + 2 \sqrt{15}}{- 2}$

Step 7

Cancel the common factor of $8 + 2 \sqrt{15}$ and $- 2$ .

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Step 7.1

Factor $2$ out of $8$ .

$\frac{2 (4) + 2 \sqrt{15}}{- 2}$

Step 7.2

Factor $2$ out of $2 \sqrt{15}$ .

$\frac{2 (4) + 2 (\sqrt{15})}{- 2}$

Step 7.3

Factor $2$ out of $2 (4) + 2 (\sqrt{15})$ .

$\frac{2 (4 + \sqrt{15})}{- 2}$

Step 7.4

Move the negative one from the denominator of $\frac{4 + \sqrt{15}}{- 1}$ .

$- 1 \cdot (4 + \sqrt{15})$

Step 8

Rewrite $- 1 \cdot (4 + \sqrt{15})$ as $- (4 + \sqrt{15})$ .

$- (4 + \sqrt{15})$

Step 9

Apply the distributive property.

$- 1 \cdot 4 - \sqrt{15}$

Step 10

Multiply $- 1$ by $4$ .

$- 4 - \sqrt{15}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- 4 - \sqrt{15}$

Decimal Form:

$- 7.87298334 \dots$