Simplify (1+ square root of 2)(1+ square root of 2- square root of 3)

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

Expand $(1 + \sqrt{2}) (1 + \sqrt{2} - \sqrt{3})$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $\sqrt{2}$ by $1$ .

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

Multiply $- \sqrt{3}$ by $1$ .

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

Multiply $\sqrt{2}$ by $1$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + \sqrt{4} + \sqrt{2} (- \sqrt{3})$

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6}$

Simplify by adding terms.

Add $1$ and $2$ .

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.64688657 \dots$