Basic Math Examples

$\frac{1 - \sqrt{3}}{3 \sqrt{5} - \sqrt{15}}$

Step 1

Multiply $\frac{1 - \sqrt{3}}{3 \sqrt{5} - \sqrt{15}}$ by $\frac{3 \sqrt{5} + \sqrt{15}}{3 \sqrt{5} + \sqrt{15}}$ .

$\frac{1 - \sqrt{3}}{3 \sqrt{5} - \sqrt{15}} \cdot \frac{3 \sqrt{5} + \sqrt{15}}{3 \sqrt{5} + \sqrt{15}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 - \sqrt{3}}{3 \sqrt{5} - \sqrt{15}}$ by $\frac{3 \sqrt{5} + \sqrt{15}}{3 \sqrt{5} + \sqrt{15}}$ .

$\frac{(1 - \sqrt{3}) (3 \sqrt{5} + \sqrt{15})}{(3 \sqrt{5} - \sqrt{15}) (3 \sqrt{5} + \sqrt{15})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(1 - \sqrt{3}) (3 \sqrt{5} + \sqrt{15})}{9 {\sqrt{5}}^{2} + 3 \sqrt{75} - 3 \sqrt{75} - {\sqrt{15}}^{2}}$

Step 2.3

Simplify.

$\frac{(1 - \sqrt{3}) (3 \sqrt{5} + \sqrt{15})}{30}$

Step 3

Simplify the numerator.

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Step 3.1

Expand $(1 - \sqrt{3}) (3 \sqrt{5} + \sqrt{15})$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$\frac{1 (3 \sqrt{5} + \sqrt{15}) - \sqrt{3} (3 \sqrt{5} + \sqrt{15})}{30}$

Step 3.1.2

Apply the distributive property.

$\frac{1 (3 \sqrt{5}) + 1 \sqrt{15} - \sqrt{3} (3 \sqrt{5} + \sqrt{15})}{30}$

Step 3.1.3

Apply the distributive property.

$\frac{1 (3 \sqrt{5}) + 1 \sqrt{15} - \sqrt{3} (3 \sqrt{5}) - \sqrt{3} \sqrt{15}}{30}$

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 $3 \sqrt{5}$ by $1$ .

$\frac{3 \sqrt{5} + 1 \sqrt{15} - \sqrt{3} (3 \sqrt{5}) - \sqrt{3} \sqrt{15}}{30}$

Step 3.2.1.2

Multiply $\sqrt{15}$ by $1$ .

$\frac{3 \sqrt{5} + \sqrt{15} - \sqrt{3} (3 \sqrt{5}) - \sqrt{3} \sqrt{15}}{30}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

Multiply $3$ by $- 1$ .

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{3} \sqrt{5} - \sqrt{3} \sqrt{15}}{30}$

Step 3.2.1.3.2

Combine using the product rule for radicals.

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{5 \cdot 3} - \sqrt{3} \sqrt{15}}{30}$

Step 3.2.1.3.3

Multiply $5$ by $3$ .

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - \sqrt{3} \sqrt{15}}{30}$

Step 3.2.1.4

Multiply $- \sqrt{3} \sqrt{15}$ .

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Step 3.2.1.4.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - \sqrt{15 \cdot 3}}{30}$

Step 3.2.1.4.2

Multiply $15$ by $3$ .

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - \sqrt{45}}{30}$

Step 3.2.1.5

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

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Step 3.2.1.5.1

Factor $9$ out of $45$ .

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - \sqrt{9 (5)}}{30}$

Step 3.2.1.5.2

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

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - \sqrt{3^{2} \cdot 5}}{30}$

Step 3.2.1.6

Pull terms out from under the radical.

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - (3 \sqrt{5})}{30}$

Step 3.2.1.7

Multiply $3$ by $- 1$ .

$\frac{3 \sqrt{5} + \sqrt{15} - 3 \sqrt{15} - 3 \sqrt{5}}{30}$

Step 3.2.2

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

$\frac{0 + \sqrt{15} - 3 \sqrt{15}}{30}$

Step 3.2.3

Add $0$ and $\sqrt{15}$ .

$\frac{\sqrt{15} - 3 \sqrt{15}}{30}$

Step 3.2.4

Subtract $3 \sqrt{15}$ from $\sqrt{15}$ .

$\frac{- 2 \sqrt{15}}{30}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $- 2$ and $30$ .

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Step 4.1.1

Factor $2$ out of $- 2 \sqrt{15}$ .

$\frac{2 (- \sqrt{15})}{30}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $30$ .

$\frac{2 (- \sqrt{15})}{2 (15)}$

Step 4.1.2.2

Cancel the common factor.

$\frac{2 (- \sqrt{15})}{2 \cdot 15}$

Step 4.1.2.3

Rewrite the expression.

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

Step 4.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25819888 \dots$