Simplify (3+ square root of 2)/(3- square root of 2)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Simplify the numerator.

Raise $3 + \sqrt{2}$ to the power of $1$ .

$\frac{{(3 + \sqrt{2})}^{1} (3 + \sqrt{2})}{7}$

Raise $3 + \sqrt{2}$ to the power of $1$ .

$\frac{{(3 + \sqrt{2})}^{1} {(3 + \sqrt{2})}^{1}}{7}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(3 + \sqrt{2})}^{1 + 1}}{7}$

Add $1$ and $1$ .

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

Step 4

Rewrite ${(3 + \sqrt{2})}^{2}$ as $(3 + \sqrt{2}) (3 + \sqrt{2})$ .

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

$\frac{9 + 3 \sqrt{2} + 3 \sqrt{2} + \sqrt{4}}{7}$

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

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

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

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

Add $9$ and $2$ .

$\frac{11 + 3 \sqrt{2} + 3 \sqrt{2}}{7}$

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

$\frac{11 + 6 \sqrt{2}}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{11 + 6 \sqrt{2}}{7}$

Decimal Form:

$2.78361162 \dots$

Simplify (3+ square root of 2)/(3- square root of 2)

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