Algebra Examples

$\frac{3 \sqrt{2}}{2 - \sqrt{7}}$

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Cancel the common factor of $3$ and $- 3$ .

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Step 2.4.1

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

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

Step 2.4.2

Move the negative one from the denominator of $\frac{\sqrt{2} (2 + \sqrt{7})}{- 1}$ .

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

Step 2.5

Rewrite $- 1 \cdot (\sqrt{2} (2 + \sqrt{7}))$ as $- (\sqrt{2} (2 + \sqrt{7}))$ .

$- (\sqrt{2} (2 + \sqrt{7}))$

Step 2.6

Apply the distributive property.

$- (\sqrt{2} \cdot 2 + \sqrt{2} \sqrt{7})$

Step 2.7

Move $2$ to the left of $\sqrt{2}$ .

$- (2 \cdot \sqrt{2} + \sqrt{2} \sqrt{7})$

Step 2.8

Combine using the product rule for radicals.

$- (2 \cdot \sqrt{2} + \sqrt{2 \cdot 7})$

Step 3

Multiply $2$ by $7$ .

$- (2 \sqrt{2} + \sqrt{14})$

Step 4

Simplify by multiplying through.

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Step 4.1

Apply the distributive property.

$- (2 \sqrt{2}) - \sqrt{14}$

Step 4.2

Multiply $2$ by $- 1$ .

$- 2 \sqrt{2} - \sqrt{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{2} - \sqrt{14}$

Decimal Form:

$- 6.57008451 \dots$