Basic Math Examples

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

Step 1

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

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

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(2 + \sqrt{2}) (3 - \sqrt{3})}{6}$

Step 3

Expand $(2 + \sqrt{2}) (3 - \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{2 (3 - \sqrt{3}) + \sqrt{2} (3 - \sqrt{3})}{6}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- \sqrt{3}) + \sqrt{2} (3 - \sqrt{3})}{6}$

Step 3.3

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- \sqrt{3}) + \sqrt{2} \cdot 3 + \sqrt{2} (- \sqrt{3})}{6}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply $2$ by $3$ .

$\frac{6 + 2 (- \sqrt{3}) + \sqrt{2} \cdot 3 + \sqrt{2} (- \sqrt{3})}{6}$

Step 4.2

Multiply $- 1$ by $2$ .

$\frac{6 - 2 \sqrt{3} + \sqrt{2} \cdot 3 + \sqrt{2} (- \sqrt{3})}{6}$

Step 4.3

Move $3$ to the left of $\sqrt{2}$ .

$\frac{6 - 2 \sqrt{3} + 3 \cdot \sqrt{2} + \sqrt{2} (- \sqrt{3})}{6}$

Step 4.4

Multiply $\sqrt{2} (- \sqrt{3})$ .

Tap for more steps...

Step 4.4.1

Combine using the product rule for radicals.

$\frac{6 - 2 \sqrt{3} + 3 \sqrt{2} - \sqrt{2 \cdot 3}}{6}$

Step 4.4.2

Multiply $2$ by $3$ .

$\frac{6 - 2 \sqrt{3} + 3 \sqrt{2} - \sqrt{6}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{6 - 2 \sqrt{3} + 3 \sqrt{2} - \sqrt{6}}{6}$

Decimal Form:

$0.72150822 \dots$