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

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 3

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 4

Simplify.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify each term.

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Step 6.1

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 6.2

Multiply $6$ by $5$ .

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

Step 6.3

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

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Step 6.3.1

Combine using the product rule for radicals.

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

Step 6.3.2

Multiply $6$ by $3$ .

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

Step 6.4

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

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Step 6.4.1

Factor $9$ out of $18$ .

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

Step 6.4.2

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

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

Step 6.5

Pull terms out from under the radical.

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

Step 6.6

Multiply $3$ by $- 1$ .

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

Step 6.7

Combine using the product rule for radicals.

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

Step 6.8

Multiply $11$ by $5$ .

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

Step 6.9

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

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Step 6.9.1

Combine using the product rule for radicals.

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

Step 6.9.2

Multiply $11$ by $3$ .

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

Step 7

Split the fraction $\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} - \sqrt{33}}{2}$ into two fractions.

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

Step 8

Split the fraction $\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55}}{2}$ into two fractions.

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

Step 9

Split the fraction $\frac{\sqrt{30} - 3 \sqrt{2}}{2}$ into two fractions.

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

Step 10

Move the negative in front of the fraction.

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

Step 11

Move the negative in front of the fraction.

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.45311036 \dots$