Simplify (4+ square root of 2)/(2- square root of 2)

$\frac{4 + \sqrt{2}}{2 - \sqrt{2}}$

Multiply $\frac{4 + \sqrt{2}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

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

Combine fractions.

Multiply $\frac{4 + \sqrt{2}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$\frac{(4 + \sqrt{2}) (2 + \sqrt{2})}{(2 - \sqrt{2}) (2 + \sqrt{2})}$

Expand the denominator using the FOIL method.

$\frac{(4 + \sqrt{2}) (2 + \sqrt{2})}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}$

Simplify.

$\frac{(4 + \sqrt{2}) (2 + \sqrt{2})}{2}$

Expand $(4 + \sqrt{2}) (2 + \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (2 + \sqrt{2}) + \sqrt{2} (2 + \sqrt{2})}{2}$

Apply the distributive property.

$\frac{4 \cdot 2 + 4 \sqrt{2} + \sqrt{2} (2 + \sqrt{2})}{2}$

Apply the distributive property.

$\frac{4 \cdot 2 + 4 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $2$ .

$\frac{8 + 4 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{2}$

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

$\frac{8 + 4 \sqrt{2} + 2 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{2}$

Combine using the product rule for radicals.

$\frac{8 + 4 \sqrt{2} + 2 \sqrt{2} + \sqrt{2 \cdot 2}}{2}$

Multiply $2$ by $2$ .

$\frac{8 + 4 \sqrt{2} + 2 \sqrt{2} + \sqrt{4}}{2}$

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

$\frac{8 + 4 \sqrt{2} + 2 \sqrt{2} + \sqrt{2^{2}}}{2}$

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

$\frac{8 + 4 \sqrt{2} + 2 \sqrt{2} + 2}{2}$

Add $8$ and $2$ .

$\frac{10 + 4 \sqrt{2} + 2 \sqrt{2}}{2}$

Add $4 \sqrt{2}$ and $2 \sqrt{2}$ .

$\frac{10 + 6 \sqrt{2}}{2}$

Cancel the common factor of $10 + 6 \sqrt{2}$ and $2$ .

Factor $2$ out of $10$ .

$\frac{2 \cdot 5 + 6 \sqrt{2}}{2}$

Factor $2$ out of $6 \sqrt{2}$ .

$\frac{2 \cdot 5 + 2 (3 \sqrt{2})}{2}$

Factor $2$ out of $2 (5) + 2 (3 \sqrt{2})$ .

$\frac{2 (5 + 3 \sqrt{2})}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (5 + 3 \sqrt{2})}{2 (1)}$

Cancel the common factor.

$\frac{2 (5 + 3 \sqrt{2})}{2 \cdot 1}$

Rewrite the expression.

$\frac{5 + 3 \sqrt{2}}{1}$

Divide $5 + 3 \sqrt{2}$ by $1$ .

$5 + 3 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$5 + 3 \sqrt{2}$

Decimal Form:

$9.24264068 \dots$

Simplify (4+ square root of 2)/(2- square root of 2)

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