Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Cancel the common factor of $3 \sqrt{2} - 3$ and $9$ .

Tap for more steps...

Step 2.4.1

Factor $3$ out of $(\sqrt{2} + 3) (3 \sqrt{2} - 3)$ .

$\frac{3 ((\sqrt{2} + 3) (\sqrt{2} - 1))}{9}$

Step 2.4.2

Cancel the common factors.

Tap for more steps...

Step 2.4.2.1

Factor $3$ out of $9$ .

$\frac{3 ((\sqrt{2} + 3) (\sqrt{2} - 1))}{3 (3)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{3 ((\sqrt{2} + 3) (\sqrt{2} - 1))}{3 \cdot 3}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(\sqrt{2} + 3) (\sqrt{2} - 1)}{3}$

Step 3

Expand $(\sqrt{2} + 3) (\sqrt{2} - 1)$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{\sqrt{2} (\sqrt{2} - 1) + 3 (\sqrt{2} - 1)}{3}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2} + \sqrt{2} \cdot - 1 + 3 (\sqrt{2} - 1)}{3}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2} + \sqrt{2} \cdot - 1 + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 2} + \sqrt{2} \cdot - 1 + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4.1.2

Multiply $2$ by $2$ .

$\frac{\sqrt{4} + \sqrt{2} \cdot - 1 + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4.1.3

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}} + \sqrt{2} \cdot - 1 + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4.1.4

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

$\frac{2 + \sqrt{2} \cdot - 1 + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4.1.5

Move $- 1$ to the left of $\sqrt{2}$ .

$\frac{2 - 1 \cdot \sqrt{2} + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4.1.6

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$\frac{2 - \sqrt{2} + 3 \sqrt{2} + 3 \cdot - 1}{3}$

Step 4.1.7

Multiply $3$ by $- 1$ .

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

Step 4.2

Subtract $3$ from $2$ .

$\frac{- 1 - \sqrt{2} + 3 \sqrt{2}}{3}$

Step 4.3

Add $- \sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{- 1 + 2 \sqrt{2}}{3}$

Step 5

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) + 2 \sqrt{2}}{3}$

Step 6

Factor $- 1$ out of $2 \sqrt{2}$ .

$\frac{- 1 (1) - (- 2 \sqrt{2})}{3}$

Step 7

Factor $- 1$ out of $- 1 (1) - (- 2 \sqrt{2})$ .

$\frac{- 1 (1 - 2 \sqrt{2})}{3}$

Step 8

Move the negative in front of the fraction.

$- \frac{1 - 2 \sqrt{2}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{1 - 2 \sqrt{2}}{3}$

Decimal Form:

$0.60947570 \dots$