Basic Math Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 + \sqrt{3}) (3 + \sqrt{7})}{9 + 3 \sqrt{7} - 3 \sqrt{7} - {\sqrt{7}}^{2}}$

Step 2.3

Simplify.

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

Step 3

Expand $(3 + \sqrt{3}) (3 + \sqrt{7})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{7} + \sqrt{3} (3 + \sqrt{7})}{2}$

Step 3.3

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{7} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{7}}{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{9 + 3 \sqrt{7} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{7}}{2}$

Step 4.2

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

$\frac{9 + 3 \sqrt{7} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{7}}{2}$

Step 4.3

Combine using the product rule for radicals.

$\frac{9 + 3 \sqrt{7} + 3 \sqrt{3} + \sqrt{3 \cdot 7}}{2}$

Step 4.4

Multiply $3$ by $7$ .

$\frac{9 + 3 \sqrt{7} + 3 \sqrt{3} + \sqrt{21}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{9 + 3 \sqrt{7} + 3 \sqrt{3} + \sqrt{21}}{2}$

Decimal Form:

$13.35799102 \dots$