Algebra Examples

$\frac{3 \sqrt{11} + \sqrt{3}}{\sqrt{11} - \sqrt{3}}$

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 \sqrt{11} + \sqrt{3}) (\sqrt{11} + \sqrt{3})}{{\sqrt{11}}^{2} + \sqrt{33} - \sqrt{33} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(3 \sqrt{11} + \sqrt{3}) (\sqrt{11} + \sqrt{3})}{8}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{3 \sqrt{11} (\sqrt{11} + \sqrt{3}) + \sqrt{3} (\sqrt{11} + \sqrt{3})}{8}$

Step 3.2

Apply the distributive property.

$\frac{3 \sqrt{11} \sqrt{11} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} (\sqrt{11} + \sqrt{3})}{8}$

Step 3.3

Apply the distributive property.

$\frac{3 \sqrt{11} \sqrt{11} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $3 \sqrt{11} \sqrt{11}$ .

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Step 4.1.1.1

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{3 ({\sqrt{11}}^{1} \sqrt{11}) + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.1.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{3 ({\sqrt{11}}^{1} {\sqrt{11}}^{1}) + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 {\sqrt{11}}^{1 + 1} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.1.4

Add $1$ and $1$ .

$\frac{3 {\sqrt{11}}^{2} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{3 {(11^{\frac{1}{2}})}^{2} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \cdot 11^{\frac{1}{2} \cdot 2} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \cdot 11^{\frac{2}{2}} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

$\frac{3 \cdot 11^{\frac{2}{2}} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.2.4.2

Rewrite the expression.

$\frac{3 \cdot 11^{1} + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.2.5

Evaluate the exponent.

$\frac{3 \cdot 11 + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.3

Multiply $3$ by $11$ .

$\frac{33 + 3 \sqrt{11} \sqrt{3} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.4

Multiply $3 \sqrt{11} \sqrt{3}$ .

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Step 4.1.4.1

Combine using the product rule for radicals.

$\frac{33 + 3 \sqrt{3 \cdot 11} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.4.2

Multiply $3$ by $11$ .

$\frac{33 + 3 \sqrt{33} + \sqrt{3} \sqrt{11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.5

Combine using the product rule for radicals.

$\frac{33 + 3 \sqrt{33} + \sqrt{3 \cdot 11} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.6

Multiply $3$ by $11$ .

$\frac{33 + 3 \sqrt{33} + \sqrt{33} + \sqrt{3} \sqrt{3}}{8}$

Step 4.1.7

Combine using the product rule for radicals.

$\frac{33 + 3 \sqrt{33} + \sqrt{33} + \sqrt{3 \cdot 3}}{8}$

Step 4.1.8

Multiply $3$ by $3$ .

$\frac{33 + 3 \sqrt{33} + \sqrt{33} + \sqrt{9}}{8}$

Step 4.1.9

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

$\frac{33 + 3 \sqrt{33} + \sqrt{33} + \sqrt{3^{2}}}{8}$

Step 4.1.10

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

$\frac{33 + 3 \sqrt{33} + \sqrt{33} + 3}{8}$

Step 4.2

Add $33$ and $3$ .

$\frac{36 + 3 \sqrt{33} + \sqrt{33}}{8}$

Step 4.3

Add $3 \sqrt{33}$ and $\sqrt{33}$ .

$\frac{36 + 4 \sqrt{33}}{8}$

Step 5

Cancel the common factor of $36 + 4 \sqrt{33}$ and $8$ .

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Step 5.1

Factor $4$ out of $36$ .

$\frac{4 \cdot 9 + 4 \sqrt{33}}{8}$

Step 5.2

Factor $4$ out of $4 \sqrt{33}$ .

$\frac{4 \cdot 9 + 4 (\sqrt{33})}{8}$

Step 5.3

Factor $4$ out of $4 \cdot 9 + 4 (\sqrt{33})$ .

$\frac{4 \cdot (9 + \sqrt{33})}{8}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $4$ out of $8$ .

$\frac{4 \cdot (9 + \sqrt{33})}{4 (2)}$

Step 5.4.2

Cancel the common factor.

$\frac{4 \cdot (9 + \sqrt{33})}{4 \cdot 2}$

Step 5.4.3

Rewrite the expression.

$\frac{9 + \sqrt{33}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9 + \sqrt{33}}{2}$

Decimal Form:

$7.37228132 \dots$