Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(4 \sqrt{2} - 4) (\sqrt{6} + \sqrt{3})}{{\sqrt{6}}^{2} + \sqrt{18} - \sqrt{18} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Expand $(4 \sqrt{2} - 4) (\sqrt{6} + \sqrt{3})$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $4 \sqrt{2} \sqrt{6}$ .

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Step 3.2.1.1.1

Combine using the product rule for radicals.

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

Step 3.2.1.1.2

Multiply $6$ by $2$ .

$\frac{4 \sqrt{12} + 4 \sqrt{2} \sqrt{3} - 4 \sqrt{6} - 4 \sqrt{3}}{3}$

Step 3.2.1.2

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.2.1.2.1

Factor $4$ out of $12$ .

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

Step 3.2.1.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.1.3

Pull terms out from under the radical.

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

Step 3.2.1.4

Multiply $2$ by $4$ .

$\frac{8 \sqrt{3} + 4 \sqrt{2} \sqrt{3} - 4 \sqrt{6} - 4 \sqrt{3}}{3}$

Step 3.2.1.5

Multiply $4 \sqrt{2} \sqrt{3}$ .

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Step 3.2.1.5.1

Combine using the product rule for radicals.

$\frac{8 \sqrt{3} + 4 \sqrt{3 \cdot 2} - 4 \sqrt{6} - 4 \sqrt{3}}{3}$

Step 3.2.1.5.2

Multiply $3$ by $2$ .

$\frac{8 \sqrt{3} + 4 \sqrt{6} - 4 \sqrt{6} - 4 \sqrt{3}}{3}$

Step 3.2.2

Subtract $4 \sqrt{3}$ from $8 \sqrt{3}$ .

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

Step 3.2.3

Subtract $4 \sqrt{6}$ from $4 \sqrt{6}$ .

$\frac{4 \sqrt{3} + 0}{3}$

Step 3.2.4

Add $4 \sqrt{3}$ and $0$ .

$\frac{4 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \sqrt{3}}{3}$

Decimal Form:

$2.30940107 \dots$