Basic Math Examples

\frac{\sqrt{2}}{9 - \sqrt{3}}

$\frac{\sqrt{2}}{9 - \sqrt{3}}$

Step 1

Multiply

\frac{\sqrt{2}}{9 - \sqrt{3}}

$\frac{\sqrt{2}}{9 - \sqrt{3}}$ by

\frac{9 + \sqrt{3}}{9 + \sqrt{3}}

$\frac{9 + \sqrt{3}}{9 + \sqrt{3}}$ .

\frac{\sqrt{2}}{9 - \sqrt{3}} \cdot \frac{9 + \sqrt{3}}{9 + \sqrt{3}}

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

Step 2

Simplify terms.

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Step 2.1

Multiply

\frac{\sqrt{2}}{9 - \sqrt{3}}

$\frac{\sqrt{2}}{9 - \sqrt{3}}$ by

\frac{9 + \sqrt{3}}{9 + \sqrt{3}}

$\frac{9 + \sqrt{3}}{9 + \sqrt{3}}$ .

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

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

Step 2.2

Expand the denominator using the FOIL method.

\frac{\sqrt{2} (9 + \sqrt{3})}{81 + 9 \sqrt{3} - 9 \sqrt{3} - {\sqrt{3}}^{2}}

$\frac{\sqrt{2} (9 + \sqrt{3})}{81 + 9 \sqrt{3} - 9 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

\frac{\sqrt{2} (9 + \sqrt{3})}{78}

$\frac{\sqrt{2} (9 + \sqrt{3})}{78}$

Step 2.4

Apply the distributive property.

$\frac{\sqrt{2} \cdot 9 + \sqrt{2} \sqrt{3}}{78}$

Step 2.5

Move

$9$ to the left of

$\sqrt{2}$ .

$\frac{9 \cdot \sqrt{2} + \sqrt{2} \sqrt{3}}{78}$

Step 2.6

Combine using the product rule for radicals.

$\frac{9 \cdot \sqrt{2} + \sqrt{2 \cdot 3}}{78}$

Step 3

Multiply

$2$ by

$3$ .

$\frac{9 \sqrt{2} + \sqrt{6}}{78}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{9 \sqrt{2} + \sqrt{6}}{78}$

Decimal Form:

$0.19458220 \dots$