Algebra Examples

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

Step 1

Multiply $\frac{6 \sqrt{3} - 1}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{6 \sqrt{3} - 1}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(6 \sqrt{3} - 1) (\sqrt{2} - \sqrt{6})}{{\sqrt{2}}^{2} - \sqrt{12} + \sqrt{12} - {\sqrt{6}}^{2}}$

Step 2.3

Simplify.

$\frac{(6 \sqrt{3} - 1) (\sqrt{2} - \sqrt{6})}{- 4}$

Step 3

Expand $(6 \sqrt{3} - 1) (\sqrt{2} - \sqrt{6})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{6 \sqrt{3} (\sqrt{2} - \sqrt{6}) - 1 (\sqrt{2} - \sqrt{6})}{- 4}$

Step 3.2

Apply the distributive property.

$\frac{6 \sqrt{3} \sqrt{2} + 6 \sqrt{3} (- \sqrt{6}) - 1 (\sqrt{2} - \sqrt{6})}{- 4}$

Step 3.3

Apply the distributive property.

$\frac{6 \sqrt{3} \sqrt{2} + 6 \sqrt{3} (- \sqrt{6}) - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

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 $6 \sqrt{3} \sqrt{2}$ .

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Step 4.1.1.1

Combine using the product rule for radicals.

$\frac{6 \sqrt{2 \cdot 3} + 6 \sqrt{3} (- \sqrt{6}) - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.1.2

Multiply $2$ by $3$ .

$\frac{6 \sqrt{6} + 6 \sqrt{3} (- \sqrt{6}) - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 1$ by $6$ .

$\frac{6 \sqrt{6} - 6 \sqrt{3} \sqrt{6} - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.2.2

Combine using the product rule for radicals.

$\frac{6 \sqrt{6} - 6 \sqrt{6 \cdot 3} - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.2.3

Multiply $6$ by $3$ .

$\frac{6 \sqrt{6} - 6 \sqrt{18} - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.3

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

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Step 4.1.3.1

Factor $9$ out of $18$ .

$\frac{6 \sqrt{6} - 6 \sqrt{9 (2)} - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.3.2

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

$\frac{6 \sqrt{6} - 6 \sqrt{3^{2} \cdot 2} - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.4

Pull terms out from under the radical.

$\frac{6 \sqrt{6} - 6 (3 \sqrt{2}) - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.5

Multiply $3$ by $- 6$ .

$\frac{6 \sqrt{6} - 18 \sqrt{2} - 1 \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.6

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$\frac{6 \sqrt{6} - 18 \sqrt{2} - \sqrt{2} - 1 (- \sqrt{6})}{- 4}$

Step 4.1.7

Multiply $- 1 (- \sqrt{6})$ .

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Step 4.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{6 \sqrt{6} - 18 \sqrt{2} - \sqrt{2} + 1 \sqrt{6}}{- 4}$

Step 4.1.7.2

Multiply $\sqrt{6}$ by $1$ .

$\frac{6 \sqrt{6} - 18 \sqrt{2} - \sqrt{2} + \sqrt{6}}{- 4}$

Step 4.2

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

$\frac{7 \sqrt{6} - 18 \sqrt{2} - \sqrt{2}}{- 4}$

Step 4.3

Subtract $\sqrt{2}$ from $- 18 \sqrt{2}$ .

$\frac{7 \sqrt{6} - 19 \sqrt{2}}{- 4}$

Step 5

Move the negative in front of the fraction.

$- \frac{7 \sqrt{6} - 19 \sqrt{2}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{7 \sqrt{6} - 19 \sqrt{2}}{4}$

Decimal Form:

$2.43090737 \dots$