Algebra Examples

$\frac{9}{\sqrt{5} - \sqrt{8}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\frac{9}{\sqrt{5} - \sqrt{4 (2)}}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{9}{\sqrt{5} - \sqrt{2^{2} \cdot 2}}$

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{9}{\sqrt{5} - 2 \sqrt{2}}$

Step 2

Multiply $\frac{9}{\sqrt{5} - 2 \sqrt{2}}$ by $\frac{\sqrt{5} + 2 \sqrt{2}}{\sqrt{5} + 2 \sqrt{2}}$ .

$\frac{9}{\sqrt{5} - 2 \sqrt{2}} \cdot \frac{\sqrt{5} + 2 \sqrt{2}}{\sqrt{5} + 2 \sqrt{2}}$

Step 3

Multiply $\frac{9}{\sqrt{5} - 2 \sqrt{2}}$ by $\frac{\sqrt{5} + 2 \sqrt{2}}{\sqrt{5} + 2 \sqrt{2}}$ .

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

Step 4

Expand the denominator using the FOIL method.

$\frac{9 (\sqrt{5} + 2 \sqrt{2})}{{\sqrt{5}}^{2} + 2 \sqrt{10} - 2 \sqrt{10} - 4 {\sqrt{2}}^{2}}$

Step 5

Simplify.

$\frac{9 (\sqrt{5} + 2 \sqrt{2})}{- 3}$

Step 6

Cancel the common factor of $9$ and $- 3$ .

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Step 6.1

Factor $3$ out of $9 (\sqrt{5} + 2 \sqrt{2})$ .

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

Step 6.2

Move the negative one from the denominator of $\frac{3 (\sqrt{5} + 2 \sqrt{2})}{- 1}$ .

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

Step 7

Rewrite $- 1 \cdot (3 (\sqrt{5} + 2 \sqrt{2}))$ as $- (3 (\sqrt{5} + 2 \sqrt{2}))$ .

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $2$ by $3$ .

$- (3 \sqrt{5} + 6 \sqrt{2})$

Step 10

Apply the distributive property.

$- (3 \sqrt{5}) - (6 \sqrt{2})$

Step 11

Multiply.

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Step 11.1

Multiply $3$ by $- 1$ .

$- 3 \sqrt{5} - (6 \sqrt{2})$

Step 11.2

Multiply $6$ by $- 1$ .

$- 3 \sqrt{5} - 6 \sqrt{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- 3 \sqrt{5} - 6 \sqrt{2}$

Decimal Form:

$- 15.19348530 \dots$