Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{\sqrt{5} (\sqrt{5} + \sqrt{15})}{- 10}$

Step 2.4

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{15}}{- 10}$

Step 2.5

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 5} + \sqrt{5} \sqrt{15}}{- 10}$

Step 2.6

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 5} + \sqrt{5 \cdot 15}}{- 10}$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Multiply $5$ by $5$ .

$\frac{\sqrt{25} + \sqrt{5 \cdot 15}}{- 10}$

Step 3.2

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

$\frac{\sqrt{5^{2}} + \sqrt{5 \cdot 15}}{- 10}$

Step 3.3

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

$\frac{5 + \sqrt{5 \cdot 15}}{- 10}$

Step 3.4

Multiply $5$ by $15$ .

$\frac{5 + \sqrt{75}}{- 10}$

Step 3.5

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

Tap for more steps...

Step 3.5.1

Factor $25$ out of $75$ .

$\frac{5 + \sqrt{25 (3)}}{- 10}$

Step 3.5.2

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

$\frac{5 + \sqrt{5^{2} \cdot 3}}{- 10}$

Step 3.6

Pull terms out from under the radical.

$\frac{5 + 5 \sqrt{3}}{- 10}$

Step 4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.1

Cancel the common factor of $5 + 5 \sqrt{3}$ and $- 10$ .

Tap for more steps...

Step 4.1.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 1 + 5 \sqrt{3}}{- 10}$

Step 4.1.2

Factor $5$ out of $5 \sqrt{3}$ .

$\frac{5 \cdot 1 + 5 (\sqrt{3})}{- 10}$

Step 4.1.3

Factor $5$ out of $5 \cdot 1 + 5 (\sqrt{3})$ .

$\frac{5 \cdot (1 + \sqrt{3})}{- 10}$

Step 4.1.4

Cancel the common factors.

Tap for more steps...

Step 4.1.4.1

Factor $5$ out of $- 10$ .

$\frac{5 \cdot (1 + \sqrt{3})}{5 (- 2)}$

Step 4.1.4.2

Cancel the common factor.

$\frac{5 \cdot (1 + \sqrt{3})}{5 \cdot - 2}$

Step 4.1.4.3

Rewrite the expression.

$\frac{1 + \sqrt{3}}{- 2}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{1 + \sqrt{3}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1 + \sqrt{3}}{2}$

Decimal Form:

$- 1.36602540 \dots$