Finite Math Examples

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

Step 1

Multiply $\frac{\sqrt{15} - 3}{\sqrt{5} - \sqrt{3}}$ by $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{15} - 3}{\sqrt{5} - \sqrt{3}}$ by $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ .

$\frac{(\sqrt{15} - 3) (\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Expand $(\sqrt{15} - 3) (\sqrt{5} + \sqrt{3})$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

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

Combine using the product rule for radicals.

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

Step 3.2.1.2

Multiply $15$ by $5$ .

$\frac{\sqrt{75} + \sqrt{15} \sqrt{3} - 3 \sqrt{5} - 3 \sqrt{3}}{2}$

Step 3.2.1.3

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.2.1.3.1

Factor $25$ out of $75$ .

$\frac{\sqrt{25 (3)} + \sqrt{15} \sqrt{3} - 3 \sqrt{5} - 3 \sqrt{3}}{2}$

Step 3.2.1.3.2

Rewrite $25$ as $5^{2}$ .

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

Step 3.2.1.4

Pull terms out from under the radical.

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

Step 3.2.1.5

Combine using the product rule for radicals.

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

Step 3.2.1.6

Multiply $15$ by $3$ .

$\frac{5 \sqrt{3} + \sqrt{45} - 3 \sqrt{5} - 3 \sqrt{3}}{2}$

Step 3.2.1.7

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 3.2.1.7.1

Factor $9$ out of $45$ .

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

Step 3.2.1.7.2

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

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

Step 3.2.1.8

Pull terms out from under the radical.

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

Step 3.2.2

Subtract $3 \sqrt{3}$ from $5 \sqrt{3}$ .

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

Step 3.2.3

Subtract $3 \sqrt{5}$ from $3 \sqrt{5}$ .

$\frac{2 \sqrt{3} + 0}{2}$

Step 3.2.4

Add $2 \sqrt{3}$ and $0$ .

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Divide $\sqrt{3}$ by $1$ .

$\sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3}$

Decimal Form:

$1.73205080 \dots$