Algebra Examples

$\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}$ by $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ .

$\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}} \cdot \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}$ by $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ .

$\frac{(\sqrt{6} + \sqrt{11}) (\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{6} + \sqrt{11}) (\sqrt{5} - \sqrt{3})}{{\sqrt{5}}^{2} - \sqrt{15} + \sqrt{15} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{6} + \sqrt{11}) (\sqrt{5} - \sqrt{3})}{2}$

Step 3

Expand $(\sqrt{6} + \sqrt{11}) (\sqrt{5} - \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{6} (\sqrt{5} - \sqrt{3}) + \sqrt{11} (\sqrt{5} - \sqrt{3})}{2}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{5} + \sqrt{6} (- \sqrt{3}) + \sqrt{11} (\sqrt{5} - \sqrt{3})}{2}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{5} + \sqrt{6} (- \sqrt{3}) + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4

Simplify each term.

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Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 5} + \sqrt{6} (- \sqrt{3}) + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.2

Multiply $6$ by $5$ .

$\frac{\sqrt{30} + \sqrt{6} (- \sqrt{3}) + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.3

Multiply $\sqrt{6} (- \sqrt{3})$ .

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Step 4.3.1

Combine using the product rule for radicals.

$\frac{\sqrt{30} - \sqrt{6 \cdot 3} + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.3.2

Multiply $6$ by $3$ .

$\frac{\sqrt{30} - \sqrt{18} + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.4.1

Factor $9$ out of $18$ .

$\frac{\sqrt{30} - \sqrt{9 (2)} + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.4.2

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

$\frac{\sqrt{30} - \sqrt{3^{2} \cdot 2} + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.5

Pull terms out from under the radical.

$\frac{\sqrt{30} - (3 \sqrt{2}) + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.6

Multiply $3$ by $- 1$ .

$\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.7

Combine using the product rule for radicals.

$\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{11 \cdot 5} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.8

Multiply $11$ by $5$ .

$\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} + \sqrt{11} (- \sqrt{3})}{2}$

Step 4.9

Multiply $\sqrt{11} (- \sqrt{3})$ .

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Step 4.9.1

Combine using the product rule for radicals.

$\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} - \sqrt{11 \cdot 3}}{2}$

Step 4.9.2

Multiply $11$ by $3$ .

$\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} - \sqrt{33}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} - \sqrt{33}}{2}$

Decimal Form:

$1.45311036 \dots$