Algebra Examples

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

Step 1

Multiply $\frac{\sqrt{3}}{5 - \sqrt{2}}$ by $\frac{5 + \sqrt{2}}{5 + \sqrt{2}}$ .

$\frac{\sqrt{3}}{5 - \sqrt{2}} \cdot \frac{5 + \sqrt{2}}{5 + \sqrt{2}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{\sqrt{3}}{5 - \sqrt{2}}$ by $\frac{5 + \sqrt{2}}{5 + \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{\sqrt{3} (5 + \sqrt{2})}{25 + 5 \sqrt{2} - 5 \sqrt{2} - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{\sqrt{3} (5 + \sqrt{2})}{23}$

Step 2.4

Apply the distributive property.

$\frac{\sqrt{3} \cdot 5 + \sqrt{3} \sqrt{2}}{23}$

Step 2.5

Move $5$ to the left of $\sqrt{3}$ .

$\frac{5 \cdot \sqrt{3} + \sqrt{3} \sqrt{2}}{23}$

Step 2.6

Combine using the product rule for radicals.

$\frac{5 \cdot \sqrt{3} + \sqrt{3 \cdot 2}}{23}$

Step 3

Multiply $3$ by $2$ .

$\frac{5 \sqrt{3} + \sqrt{6}}{23}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{3} + \sqrt{6}}{23}$

Decimal Form:

$0.48303233 \dots$