Algebra Examples

$\frac{\sqrt{10}}{\sqrt{5} - 2}$

Step 1

Multiply $\frac{\sqrt{10}}{\sqrt{5} - 2}$ by $\frac{\sqrt{5} + 2}{\sqrt{5} + 2}$ .

$\frac{\sqrt{10}}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{\sqrt{10}}{\sqrt{5} - 2}$ by $\frac{\sqrt{5} + 2}{\sqrt{5} + 2}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{\sqrt{10} (\sqrt{5} + 2)}{1}$

Step 2.4

Divide $\sqrt{10} (\sqrt{5} + 2)$ by $1$ .

$\sqrt{10} (\sqrt{5} + 2)$

Step 2.5

Apply the distributive property.

$\sqrt{10} \sqrt{5} + \sqrt{10} \cdot 2$

Step 2.6

Combine using the product rule for radicals.

$\sqrt{10 \cdot 5} + \sqrt{10} \cdot 2$

Step 2.7

Move $2$ to the left of $\sqrt{10}$ .

$\sqrt{10 \cdot 5} + 2 \cdot \sqrt{10}$

Step 3

Simplify each term.

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Step 3.1

Multiply $10$ by $5$ .

$\sqrt{50} + 2 \cdot \sqrt{10}$

Step 3.2

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 3.2.1

Factor $25$ out of $50$ .

$\sqrt{25 (2)} + 2 \cdot \sqrt{10}$

Step 3.2.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 2} + 2 \cdot \sqrt{10}$

Step 3.3

Pull terms out from under the radical.

$5 \sqrt{2} + 2 \sqrt{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{2} + 2 \sqrt{10}$

Decimal Form:

$13.39562313 \dots$