Basic Math Examples

$\frac{\sqrt{3} - \sqrt{5}}{4 - 2 \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{3} - \sqrt{5}}{4 - 2 \sqrt{3}}$ by $\frac{4 + 2 \sqrt{3}}{4 + 2 \sqrt{3}}$ .

$\frac{\sqrt{3} - \sqrt{5}}{4 - 2 \sqrt{3}} \cdot \frac{4 + 2 \sqrt{3}}{4 + 2 \sqrt{3}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{\sqrt{3} - \sqrt{5}}{4 - 2 \sqrt{3}}$ by $\frac{4 + 2 \sqrt{3}}{4 + 2 \sqrt{3}}$ .

$\frac{(\sqrt{3} - \sqrt{5}) (4 + 2 \sqrt{3})}{(4 - 2 \sqrt{3}) (4 + 2 \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} - \sqrt{5}) (4 + 2 \sqrt{3})}{16 + 8 \sqrt{3} - 8 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{3} - \sqrt{5}) (4 + 2 \sqrt{3})}{4}$

Step 2.4

Cancel the common factor of $4 + 2 \sqrt{3}$ and $4$ .

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Step 2.4.1

Factor $2$ out of $(\sqrt{3} - \sqrt{5}) (4 + 2 \sqrt{3})$ .

$\frac{2 ((\sqrt{3} - \sqrt{5}) (2 + \sqrt{3}))}{4}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $4$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 3

Expand $(\sqrt{3} - \sqrt{5}) (2 + \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

$\frac{\sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3} - \sqrt{5} \cdot 2 - \sqrt{5} \sqrt{3}}{2}$

Step 4

Simplify each term.

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Step 4.1

Move $2$ to the left of $\sqrt{3}$ .

$\frac{2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3} - \sqrt{5} \cdot 2 - \sqrt{5} \sqrt{3}}{2}$

Step 4.2

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + \sqrt{3 \cdot 3} - \sqrt{5} \cdot 2 - \sqrt{5} \sqrt{3}}{2}$

Step 4.3

Multiply $3$ by $3$ .

$\frac{2 \sqrt{3} + \sqrt{9} - \sqrt{5} \cdot 2 - \sqrt{5} \sqrt{3}}{2}$

Step 4.4

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

$\frac{2 \sqrt{3} + \sqrt{3^{2}} - \sqrt{5} \cdot 2 - \sqrt{5} \sqrt{3}}{2}$

Step 4.5

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

$\frac{2 \sqrt{3} + 3 - \sqrt{5} \cdot 2 - \sqrt{5} \sqrt{3}}{2}$

Step 4.6

Multiply $2$ by $- 1$ .

$\frac{2 \sqrt{3} + 3 - 2 \sqrt{5} - \sqrt{5} \sqrt{3}}{2}$

Step 4.7

Multiply $- \sqrt{5} \sqrt{3}$ .

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Step 4.7.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 3 - 2 \sqrt{5} - \sqrt{3 \cdot 5}}{2}$

Step 4.7.2

Multiply $3$ by $5$ .

$\frac{2 \sqrt{3} + 3 - 2 \sqrt{5} - \sqrt{15}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3} + 3 - 2 \sqrt{5} - \sqrt{15}}{2}$

Decimal Form:

$- 0.94050884 \dots$