Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

Multiply $\sqrt{15} (- \sqrt{5})$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $15$ by $5$ .

$\frac{\sqrt{15 \cdot 3} - \sqrt{75}}{- 2}$

Step 8

Simplify each term.

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Step 8.1

Multiply $15$ by $3$ .

$\frac{\sqrt{45} - \sqrt{75}}{- 2}$

Step 8.2

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

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Step 8.2.1

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)} - \sqrt{75}}{- 2}$

Step 8.2.2

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

$\frac{\sqrt{3^{2} \cdot 5} - \sqrt{75}}{- 2}$

Step 8.3

Pull terms out from under the radical.

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

Step 8.4

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

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Step 8.4.1

Factor $25$ out of $75$ .

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

Step 8.4.2

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

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

Step 8.5

Pull terms out from under the radical.

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

Step 8.6

Multiply $5$ by $- 1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.97602505 \dots$