Algebra Examples

$\frac{9}{6 - \sqrt{8}}$

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 $8$ .

$\frac{9}{6 - \sqrt{4 (2)}}$

Step 1.1.2

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

$\frac{9}{6 - \sqrt{2^{2} \cdot 2}}$

Step 1.2

Pull terms out from under the radical.

$\frac{9}{6 - (2 \sqrt{2})}$

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{9}{6 - 2 \sqrt{2}}$

Step 2

Multiply $\frac{9}{6 - 2 \sqrt{2}}$ by $\frac{6 + 2 \sqrt{2}}{6 + 2 \sqrt{2}}$ .

$\frac{9}{6 - 2 \sqrt{2}} \cdot \frac{6 + 2 \sqrt{2}}{6 + 2 \sqrt{2}}$

Step 3

Multiply $\frac{9}{6 - 2 \sqrt{2}}$ by $\frac{6 + 2 \sqrt{2}}{6 + 2 \sqrt{2}}$ .

$\frac{9 (6 + 2 \sqrt{2})}{(6 - 2 \sqrt{2}) (6 + 2 \sqrt{2})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{9 (6 + 2 \sqrt{2})}{36 + 12 \sqrt{2} - 12 \sqrt{2} - 4 {\sqrt{2}}^{2}}$

Step 5

Simplify.

$\frac{9 (6 + 2 \sqrt{2})}{28}$

Step 6

Cancel the common factor of $6 + 2 \sqrt{2}$ and $28$ .

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Step 6.1

Factor $2$ out of $9 (6 + 2 \sqrt{2})$ .

$\frac{2 (9 (3 + \sqrt{2}))}{28}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $28$ .

$\frac{2 (9 (3 + \sqrt{2}))}{2 (14)}$

Step 6.2.2

Cancel the common factor.

$\frac{2 (9 (3 + \sqrt{2}))}{2 \cdot 14}$

Step 6.2.3

Rewrite the expression.

$\frac{9 (3 + \sqrt{2})}{14}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{9 (3 + \sqrt{2})}{14}$

Decimal Form:

$2.83770871 \dots$