Linear Algebra Examples

$\frac{2 \sqrt{3 + \sqrt{6}}}{4 \sqrt{3 - \sqrt{6}}}$

Step 1

Cancel the common factor of $2$ and $4$ .

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Step 1.1

Factor $2$ out of $2 \sqrt{3 + \sqrt{6}}$ .

$\frac{2 (\sqrt{3 + \sqrt{6}})}{4 \sqrt{3 - \sqrt{6}}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $4 \sqrt{3 - \sqrt{6}}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

$\frac{\sqrt{3 + \sqrt{6}}}{2 \sqrt{3 - \sqrt{6}}}$

Step 2

Multiply $\frac{\sqrt{3 + \sqrt{6}}}{2 \sqrt{3 - \sqrt{6}}}$ by $\frac{\sqrt{3 - \sqrt{6}}}{\sqrt{3 - \sqrt{6}}}$ .

$\frac{\sqrt{3 + \sqrt{6}}}{2 \sqrt{3 - \sqrt{6}}} \cdot \frac{\sqrt{3 - \sqrt{6}}}{\sqrt{3 - \sqrt{6}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{3 + \sqrt{6}}}{2 \sqrt{3 - \sqrt{6}}}$ by $\frac{\sqrt{3 - \sqrt{6}}}{\sqrt{3 - \sqrt{6}}}$ .

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 \sqrt{3 - \sqrt{6}} \sqrt{3 - \sqrt{6}}}$

Step 3.2

Move $\sqrt{3 - \sqrt{6}}$ .

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

Step 3.3

Raise $\sqrt{3 - \sqrt{6}}$ to the power of $1$ .

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 ({\sqrt{3 - \sqrt{6}}}^{1} \sqrt{3 - \sqrt{6}})}$

Step 3.4

Raise $\sqrt{3 - \sqrt{6}}$ to the power of $1$ .

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 ({\sqrt{3 - \sqrt{6}}}^{1} {\sqrt{3 - \sqrt{6}}}^{1})}$

Step 3.5

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

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 {\sqrt{3 - \sqrt{6}}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 {\sqrt{3 - \sqrt{6}}}^{2}}$

Step 3.7

Rewrite ${\sqrt{3 - \sqrt{6}}}^{2}$ as $3 - \sqrt{6}$ .

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Step 3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - \sqrt{6}}$ as ${(3 - \sqrt{6})}^{\frac{1}{2}}$ .

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 {({(3 - \sqrt{6})}^{\frac{1}{2}})}^{2}}$

Step 3.7.2

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

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 {(3 - \sqrt{6})}^{\frac{1}{2} \cdot 2}}$

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

$\frac{\sqrt{3 + \sqrt{6}} \sqrt{3 - \sqrt{6}}}{2 {(3 - \sqrt{6})}^{1}}$

Step 3.7.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $3$ by $3$ .

$\frac{\sqrt{9 + 3 (- \sqrt{6}) + \sqrt{6} \cdot 3 + \sqrt{6} (- \sqrt{6})}}{2 (3 - \sqrt{6})}$

Step 4.3.1.2

Multiply $- 1$ by $3$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + \sqrt{6} \cdot 3 + \sqrt{6} (- \sqrt{6})}}{2 (3 - \sqrt{6})}$

Step 4.3.1.3

Move $3$ to the left of $\sqrt{6}$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \cdot \sqrt{6} + \sqrt{6} (- \sqrt{6})}}{2 (3 - \sqrt{6})}$

Step 4.3.1.4

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

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Step 4.3.1.4.1

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - ({\sqrt{6}}^{1} \sqrt{6})}}{2 (3 - \sqrt{6})}$

Step 4.3.1.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}}{2 (3 - \sqrt{6})}$

Step 4.3.1.4.3

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

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - {\sqrt{6}}^{1 + 1}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.4.4

Add $1$ and $1$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - {\sqrt{6}}^{2}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - {(6^{\frac{1}{2}})}^{2}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.5.2

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

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - 6^{\frac{1}{2} \cdot 2}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.5.3

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

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - 6^{\frac{2}{2}}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.1.5.4.1

Cancel the common factor.

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - 6^{\frac{2}{2}}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.5.4.2

Rewrite the expression.

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - 6^{1}}}{2 (3 - \sqrt{6})}$

Step 4.3.1.5.5

Evaluate the exponent.

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - 1 \cdot 6}}{2 (3 - \sqrt{6})}$

Step 4.3.1.6

Multiply $- 1$ by $6$ .

$\frac{\sqrt{9 - 3 \sqrt{6} + 3 \sqrt{6} - 6}}{2 (3 - \sqrt{6})}$

Step 4.3.2

Subtract $6$ from $9$ .

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

Step 4.3.3

Add $- 3 \sqrt{6}$ and $3 \sqrt{6}$ .

$\frac{\sqrt{3 + 0}}{2 (3 - \sqrt{6})}$

Step 4.3.4

Add $3$ and $0$ .

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

Step 5

Multiply $\frac{\sqrt{3}}{2 (3 - \sqrt{6})}$ by $\frac{3 + \sqrt{6}}{3 + \sqrt{6}}$ .

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

Step 6

Simplify terms.

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Step 6.1

Multiply $\frac{\sqrt{3}}{2 (3 - \sqrt{6})}$ by $\frac{3 + \sqrt{6}}{3 + \sqrt{6}}$ .

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $3 - \sqrt{6}$ .

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

Step 6.2.2

Expand the denominator using the FOIL method.

$\frac{\sqrt{3} (3 + \sqrt{6})}{2 (9 + 3 \sqrt{6} - 3 \sqrt{6} - {\sqrt{6}}^{2})}$

Step 6.2.3

Simplify.

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

Step 6.2.4

Multiply $2$ by $3$ .

$\frac{\sqrt{3} (3 + \sqrt{6})}{6}$

Step 6.3

Apply the distributive property.

$\frac{\sqrt{3} \cdot 3 + \sqrt{3} \sqrt{6}}{6}$

Step 6.4

Move $3$ to the left of $\sqrt{3}$ .

$\frac{3 \cdot \sqrt{3} + \sqrt{3} \sqrt{6}}{6}$

Step 6.5

Combine using the product rule for radicals.

$\frac{3 \cdot \sqrt{3} + \sqrt{3 \cdot 6}}{6}$

Step 7

Simplify each term.

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Step 7.1

Multiply $3$ by $6$ .

$\frac{3 \sqrt{3} + \sqrt{18}}{6}$

Step 7.2

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

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Step 7.2.1

Factor $9$ out of $18$ .

$\frac{3 \sqrt{3} + \sqrt{9 (2)}}{6}$

Step 7.2.2

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

$\frac{3 \sqrt{3} + \sqrt{3^{2} \cdot 2}}{6}$

Step 7.3

Pull terms out from under the radical.

$\frac{3 \sqrt{3} + 3 \sqrt{2}}{6}$

Step 8

Cancel the common factor of $3 \sqrt{3} + 3 \sqrt{2}$ and $6$ .

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Step 8.1

Factor $3$ out of $3 \sqrt{3}$ .

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

Step 8.2

Factor $3$ out of $3 \sqrt{2}$ .

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

Step 8.3

Factor $3$ out of $3 (\sqrt{3}) + 3 (\sqrt{2})$ .

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

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $3$ out of $6$ .

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.57313218 \dots$