Simplify (4+ square root of 2)/( square root of 2-3)

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

Multiply $\frac{4 + \sqrt{2}}{\sqrt{2} - 3}$ by $\frac{\sqrt{2} + 3}{\sqrt{2} + 3}$ .

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

Combine fractions.

Multiply $\frac{4 + \sqrt{2}}{\sqrt{2} - 3}$ by $\frac{\sqrt{2} + 3}{\sqrt{2} + 3}$ .

$\frac{(4 + \sqrt{2}) (\sqrt{2} + 3)}{(\sqrt{2} - 3) (\sqrt{2} + 3)}$

Expand the denominator using the FOIL method.

$\frac{(4 + \sqrt{2}) (\sqrt{2} + 3)}{{\sqrt{2}}^{2} + \sqrt{2} \cdot 3 - 3 \sqrt{2} - 9}$

Simplify.

$\frac{(4 + \sqrt{2}) (\sqrt{2} + 3)}{- 7}$

Expand $(4 + \sqrt{2}) (\sqrt{2} + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (\sqrt{2} + 3) + \sqrt{2} (\sqrt{2} + 3)}{- 7}$

Apply the distributive property.

$\frac{4 \sqrt{2} + 4 \cdot 3 + \sqrt{2} (\sqrt{2} + 3)}{- 7}$

Apply the distributive property.

$\frac{4 \sqrt{2} + 4 \cdot 3 + \sqrt{2} \sqrt{2} + \sqrt{2} \cdot 3}{- 7}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $3$ .

$\frac{4 \sqrt{2} + 12 + \sqrt{2} \sqrt{2} + \sqrt{2} \cdot 3}{- 7}$

Combine using the product rule for radicals.

$\frac{4 \sqrt{2} + 12 + \sqrt{2 \cdot 2} + \sqrt{2} \cdot 3}{- 7}$

Multiply $2$ by $2$ .

$\frac{4 \sqrt{2} + 12 + \sqrt{4} + \sqrt{2} \cdot 3}{- 7}$

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

$\frac{4 \sqrt{2} + 12 + \sqrt{2^{2}} + \sqrt{2} \cdot 3}{- 7}$

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

$\frac{4 \sqrt{2} + 12 + 2 + \sqrt{2} \cdot 3}{- 7}$

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

$\frac{4 \sqrt{2} + 12 + 2 + 3 \sqrt{2}}{- 7}$

Add $4 \sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{12 + 2 + 7 \sqrt{2}}{- 7}$

Add $12$ and $2$ .

$\frac{14 + 7 \sqrt{2}}{- 7}$

Cancel the common factor of $14 + 7 \sqrt{2}$ and $- 7$ .

Factor $7$ out of $14$ .

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

Factor $7$ out of $7 \sqrt{2}$ .

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

Factor $7$ out of $7 (2) + 7 (\sqrt{2})$ .

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

Move the negative one from the denominator of $\frac{2 + \sqrt{2}}{- 1}$ .

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

Rewrite $- 1 \cdot (2 + \sqrt{2})$ as $- (2 + \sqrt{2})$ .

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

Apply the distributive property.

$- 1 \cdot 2 - \sqrt{2}$

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{2}$

Decimal Form:

$- 3.41421356 \dots$

Simplify (4+ square root of 2)/( square root of 2-3)

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