Basic Math Examples

$\frac{6 \sqrt{2}}{3 + \sqrt{7}} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{6 \sqrt{2}}{3 + \sqrt{7}}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\frac{6 \sqrt{2}}{3 + \sqrt{7}} \cdot \frac{3 - \sqrt{7}}{3 - \sqrt{7}} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.2

Multiply $\frac{6 \sqrt{2}}{3 + \sqrt{7}}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\frac{6 \sqrt{2} (3 - \sqrt{7})}{(3 + \sqrt{7}) (3 - \sqrt{7})} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.3

Expand the denominator using the FOIL method.

$\frac{6 \sqrt{2} (3 - \sqrt{7})}{9 - 3 \sqrt{7} + \sqrt{7} \cdot 3 - {\sqrt{7}}^{2}} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.4

Simplify.

$\frac{6 \sqrt{2} (3 - \sqrt{7})}{2} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.5

Cancel the common factor of $6$ and $2$ .

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Step 1.5.1

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

$\frac{2 (3 \sqrt{2} (3 - \sqrt{7}))}{2} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $2$ .

$\frac{2 (3 \sqrt{2} (3 - \sqrt{7}))}{2 (1)} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.5.2.2

Cancel the common factor.

$\frac{2 (3 \sqrt{2} (3 - \sqrt{7}))}{2 \cdot 1} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.5.2.3

Rewrite the expression.

$\frac{3 \sqrt{2} (3 - \sqrt{7})}{1} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.5.2.4

Divide $3 \sqrt{2} (3 - \sqrt{7})$ by $1$ .

$3 \sqrt{2} (3 - \sqrt{7}) + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.6

Apply the distributive property.

$3 \sqrt{2} \cdot 3 + 3 \sqrt{2} (- \sqrt{7}) + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.7

Multiply $3$ by $3$ .

$9 \sqrt{2} + 3 \sqrt{2} (- \sqrt{7}) + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.8

Multiply $3 \sqrt{2} (- \sqrt{7})$ .

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Step 1.8.1

Multiply $- 1$ by $3$ .

$9 \sqrt{2} - 3 \sqrt{2} \sqrt{7} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.8.2

Combine using the product rule for radicals.

$9 \sqrt{2} - 3 \sqrt{7 \cdot 2} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.8.3

Multiply $7$ by $2$ .

$9 \sqrt{2} - 3 \sqrt{14} + \frac{4 \sqrt{7}}{\sqrt{2} - 1}$

Step 1.9

Multiply $\frac{4 \sqrt{7}}{\sqrt{2} - 1}$ by $\frac{\sqrt{2} + 1}{\sqrt{2} + 1}$ .

$9 \sqrt{2} - 3 \sqrt{14} + \frac{4 \sqrt{7}}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1}$

Step 1.10

Multiply $\frac{4 \sqrt{7}}{\sqrt{2} - 1}$ by $\frac{\sqrt{2} + 1}{\sqrt{2} + 1}$ .

$9 \sqrt{2} - 3 \sqrt{14} + \frac{4 \sqrt{7} (\sqrt{2} + 1)}{(\sqrt{2} - 1) (\sqrt{2} + 1)}$

Step 1.11

Expand the denominator using the FOIL method.

$9 \sqrt{2} - 3 \sqrt{14} + \frac{4 \sqrt{7} (\sqrt{2} + 1)}{{\sqrt{2}}^{2} + \sqrt{2} - \sqrt{2} - 1}$

Step 1.12

Simplify.

$9 \sqrt{2} - 3 \sqrt{14} + \frac{4 \sqrt{7} (\sqrt{2} + 1)}{1}$

Step 1.13

Divide $4 \sqrt{7} (\sqrt{2} + 1)$ by $1$ .

$9 \sqrt{2} - 3 \sqrt{14} + 4 \sqrt{7} (\sqrt{2} + 1)$

Step 1.14

Apply the distributive property.

$9 \sqrt{2} - 3 \sqrt{14} + 4 \sqrt{7} \sqrt{2} + 4 \sqrt{7} \cdot 1$

Step 1.15

Multiply $4 \sqrt{7} \sqrt{2}$ .

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Step 1.15.1

Combine using the product rule for radicals.

$9 \sqrt{2} - 3 \sqrt{14} + 4 \sqrt{2 \cdot 7} + 4 \sqrt{7} \cdot 1$

Step 1.15.2

Multiply $2$ by $7$ .

$9 \sqrt{2} - 3 \sqrt{14} + 4 \sqrt{14} + 4 \sqrt{7} \cdot 1$

Step 1.16

Multiply $4$ by $1$ .

$9 \sqrt{2} - 3 \sqrt{14} + 4 \sqrt{14} + 4 \sqrt{7}$

Step 2

Add $- 3 \sqrt{14}$ and $4 \sqrt{14}$ .

$9 \sqrt{2} + \sqrt{14} + 4 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$9 \sqrt{2} + \sqrt{14} + 4 \sqrt{7}$

Decimal Form:

$27.05258469 \dots$