Algebra Examples

$\frac{3 \sqrt{7}}{\sqrt{7} + \sqrt{3}}$

Step 1

Multiply $\frac{3 \sqrt{7}}{\sqrt{7} + \sqrt{3}}$ by $\frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ .

$\frac{3 \sqrt{7}}{\sqrt{7} + \sqrt{3}} \cdot \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}}$

Step 2

Multiply $\frac{3 \sqrt{7}}{\sqrt{7} + \sqrt{3}}$ by $\frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ .

$\frac{3 \sqrt{7} (\sqrt{7} - \sqrt{3})}{(\sqrt{7} + \sqrt{3}) (\sqrt{7} - \sqrt{3})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{3 \sqrt{7} (\sqrt{7} - \sqrt{3})}{{\sqrt{7}}^{2} - \sqrt{21} + \sqrt{21} - {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{3 \sqrt{7} (\sqrt{7} - \sqrt{3})}{4}$

Step 5

Group $\sqrt{7} - \sqrt{3}$ and $\sqrt{7}$ together.

$\frac{3 ((\sqrt{7} - \sqrt{3}) \sqrt{7})}{4}$

Step 6

Apply the distributive property.

$\frac{3 (\sqrt{7} \sqrt{7} - \sqrt{3} \sqrt{7})}{4}$

Step 7

Combine using the product rule for radicals.

$\frac{3 (\sqrt{7 \cdot 7} - \sqrt{3} \sqrt{7})}{4}$

Step 8

Multiply $- \sqrt{3} \sqrt{7}$ .

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Step 8.1

Combine using the product rule for radicals.

$\frac{3 (\sqrt{7 \cdot 7} - \sqrt{7 \cdot 3})}{4}$

Step 8.2

Multiply $7$ by $3$ .

$\frac{3 (\sqrt{7 \cdot 7} - \sqrt{21})}{4}$

Step 9

Simplify each term.

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Step 9.1

Multiply $7$ by $7$ .

$\frac{3 (\sqrt{49} - \sqrt{21})}{4}$

Step 9.2

Rewrite $49$ as $7^{2}$ .

$\frac{3 (\sqrt{7^{2}} - \sqrt{21})}{4}$

Step 9.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 (7 - \sqrt{21})}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 (7 - \sqrt{21})}{4}$

Decimal Form:

$1.81306822 \dots$