Algebra Examples

$\frac{2 \sqrt{10}}{\sqrt{5} + \sqrt{6} - \sqrt{7}}$

Step 1

Multiply $\frac{2 \sqrt{10}}{\sqrt{5} + \sqrt{6} - \sqrt{7}}$ by $\frac{\sqrt{5} + \sqrt{6} + \sqrt{7}}{\sqrt{5} + \sqrt{6} + \sqrt{7}}$ .

$\frac{2 \sqrt{10}}{\sqrt{5} + \sqrt{6} - \sqrt{7}} \cdot \frac{\sqrt{5} + \sqrt{6} + \sqrt{7}}{\sqrt{5} + \sqrt{6} + \sqrt{7}}$

Step 2

Multiply $\frac{2 \sqrt{10}}{\sqrt{5} + \sqrt{6} - \sqrt{7}}$ by $\frac{\sqrt{5} + \sqrt{6} + \sqrt{7}}{\sqrt{5} + \sqrt{6} + \sqrt{7}}$ .

$\frac{2 \sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{(\sqrt{5} + \sqrt{6} - \sqrt{7}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{2 \sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{{\sqrt{5}}^{2} + \sqrt{30} + \sqrt{35} + \sqrt{30} + {\sqrt{6}}^{2} + \sqrt{42} - \sqrt{35} - \sqrt{42} - {\sqrt{7}}^{2}}$

Step 4

Simplify.

$\frac{2 \sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{4 + 2 \sqrt{30}}$

Step 5

Cancel the common factor of $2$ and $4 + 2 \sqrt{30}$ .

Tap for more steps...

Step 5.1

Factor $2$ out of $2 \sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})$ .

$\frac{2 (\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{4 + 2 \sqrt{30}}$

Step 5.2

Cancel the common factors.

Tap for more steps...

Step 5.2.1

Factor $2$ out of $4$ .

$\frac{2 (\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{2 \cdot 2 + 2 \sqrt{30}}$

Step 5.2.2

Factor $2$ out of $2 \sqrt{30}$ .

$\frac{2 (\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{2 \cdot 2 + 2 (\sqrt{30})}$

Step 5.2.3

Factor $2$ out of $2 \cdot 2 + 2 (\sqrt{30})$ .

$\frac{2 (\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{2 \cdot (2 + \sqrt{30})}$

Step 5.2.4

Cancel the common factor.

$\frac{2 (\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{2 \cdot (2 + \sqrt{30})}$

Step 5.2.5

Rewrite the expression.

$\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{2 + \sqrt{30}}$

Step 6

Multiply $\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{2 + \sqrt{30}}$ by $\frac{2 - \sqrt{30}}{2 - \sqrt{30}}$ .

$\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{2 + \sqrt{30}} \cdot \frac{2 - \sqrt{30}}{2 - \sqrt{30}}$

Step 7

Multiply $\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{2 + \sqrt{30}}$ by $\frac{2 - \sqrt{30}}{2 - \sqrt{30}}$ .

$\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}) (2 - \sqrt{30})}{(2 + \sqrt{30}) (2 - \sqrt{30})}$

Step 8

Expand the denominator using the FOIL method.

$\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}) (2 - \sqrt{30})}{4 - 2 \sqrt{30} + \sqrt{30} \cdot 2 - {\sqrt{30}}^{2}}$

Step 9

Simplify.

$\frac{\sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7}) (2 - \sqrt{30})}{- 26}$

Step 10

Group $2 - \sqrt{30}$ and $\sqrt{10}$ together.

$\frac{(2 - \sqrt{30}) \sqrt{10} (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 11

Apply the distributive property.

$\frac{(2 \sqrt{10} - \sqrt{30} \sqrt{10}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 12

Multiply $- \sqrt{30} \sqrt{10}$ .

Tap for more steps...

Step 12.1

Combine using the product rule for radicals.

$\frac{(2 \sqrt{10} - \sqrt{10 \cdot 30}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 12.2

Multiply $10$ by $30$ .

$\frac{(2 \sqrt{10} - \sqrt{300}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 13

Simplify each term.

Tap for more steps...

Step 13.1

Rewrite $300$ as $10^{2} \cdot 3$ .

Tap for more steps...

Step 13.1.1

Factor $100$ out of $300$ .

$\frac{(2 \sqrt{10} - \sqrt{100 (3)}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 13.1.2

Rewrite $100$ as $10^{2}$ .

$\frac{(2 \sqrt{10} - \sqrt{10^{2} \cdot 3}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 13.2

Pull terms out from under the radical.

$\frac{(2 \sqrt{10} - (10 \sqrt{3})) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 13.3

Multiply $10$ by $- 1$ .

$\frac{(2 \sqrt{10} - 10 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 26}$

Step 14

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 14.1

Cancel the common factor of $2 \sqrt{10} - 10 \sqrt{3}$ and $- 26$ .

Tap for more steps...

Step 14.1.1

Factor $2$ out of $(2 \sqrt{10} - 10 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7})$ .

$\frac{2 ((\sqrt{10} - 5 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{- 26}$

Step 14.1.2

Cancel the common factors.

Tap for more steps...

Step 14.1.2.1

Factor $2$ out of $- 26$ .

$\frac{2 ((\sqrt{10} - 5 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{2 (- 13)}$

Step 14.1.2.2

Cancel the common factor.

$\frac{2 ((\sqrt{10} - 5 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7}))}{2 \cdot - 13}$

Step 14.1.2.3

Rewrite the expression.

$\frac{(\sqrt{10} - 5 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{- 13}$

Step 14.2

Move the negative in front of the fraction.

$- \frac{(\sqrt{10} - 5 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{13}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{(\sqrt{10} - 5 \sqrt{3}) (\sqrt{5} + \sqrt{6} + \sqrt{7})}{13}$

Decimal Form:

$3.10056645 \dots$