Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

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

Step 1.4

Simplify.

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

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide $(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})$ by $1$ .

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

Step 1.6

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

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Apply the distributive property.

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

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Combine using the product rule for radicals.

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

Step 1.7.1.2

Multiply $5$ by $5$ .

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

Step 1.7.1.3

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

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

Step 1.7.1.4

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

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

Step 1.7.1.5

Combine using the product rule for radicals.

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

Step 1.7.1.6

Multiply $5$ by $3$ .

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

Step 1.7.1.7

Combine using the product rule for radicals.

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

Step 1.7.1.8

Multiply $3$ by $5$ .

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

Step 1.7.1.9

Combine using the product rule for radicals.

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

Step 1.7.1.10

Multiply $3$ by $3$ .

$5 + \sqrt{15} + \sqrt{15} + \sqrt{9} - 2 \sqrt{15}$

Step 1.7.1.11

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

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

Step 1.7.1.12

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

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

Step 1.7.2

Add $5$ and $3$ .

$8 + \sqrt{15} + \sqrt{15} - 2 \sqrt{15}$

Step 1.7.3

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

$8 + 2 \sqrt{15} - 2 \sqrt{15}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $2 \sqrt{15}$ from $2 \sqrt{15}$ .

$8 + 0$

Step 2.2

Add $8$ and $0$ .

$8$