Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $1$ by $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Combine using the product rule for radicals.

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

Step 2.1.6

Multiply $2$ by $2$ .

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

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

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Step 2.1.9.1

Combine using the product rule for radicals.

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

Step 2.1.9.2

Multiply $2$ by $3$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $1$ and $2$ .

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

Step 2.2.2

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.64688657 \dots$