Algebra Examples

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

Step 1

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

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

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 $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Combine using the product rule for radicals.

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

Step 2.1.6

Multiply $2$ by $2$ .

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

Step 2.1.7

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

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

Step 2.1.8

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

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

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.

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

Step 2.1.9.2

Multiply $2$ by $3$ .

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{2} + \sqrt{3} (- \sqrt{3})$

Step 2.1.10

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

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

Step 2.1.11

Combine using the product rule for radicals.

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6} + \sqrt{3} + \sqrt{3 \cdot 2} + \sqrt{3} (- \sqrt{3})$

Step 2.1.12

Multiply $3$ by $2$ .

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

Step 2.1.13

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

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Step 2.1.13.1

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

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6} + \sqrt{3} + \sqrt{6} - ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.13.2

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

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6} + \sqrt{3} + \sqrt{6} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.13.3

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

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

Step 2.1.13.4

Add $1$ and $1$ .

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

Step 2.1.14

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6} + \sqrt{3} + \sqrt{6} - {(3^{\frac{1}{2}})}^{2}$

Step 2.1.14.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 + \sqrt{2} - \sqrt{3} + \sqrt{2} + 2 - \sqrt{6} + \sqrt{3} + \sqrt{6} - 3^{\frac{1}{2} \cdot 2}$

Step 2.1.14.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.1.14.4

Cancel the common factor of $2$ .

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Step 2.1.14.4.1

Cancel the common factor.

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

Step 2.1.14.4.2

Rewrite the expression.

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

Step 2.1.14.5

Evaluate the exponent.

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

Step 2.1.15

Multiply $- 1$ by $3$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $1$ and $2$ .

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

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract $3$ from $3$ .

$0 + \sqrt{2} - \sqrt{3} + \sqrt{2} - \sqrt{6} + \sqrt{3} + \sqrt{6}$

Step 2.2.2.2

Add $0$ and $\sqrt{2}$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

$2 \sqrt{2} + 0 - \sqrt{6} + \sqrt{6}$

Step 2.2.5

Add $2 \sqrt{2}$ and $0$ .

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

Step 2.2.6

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$2 \sqrt{2} + 0$

Step 2.2.7

Add $2 \sqrt{2}$ and $0$ .

$2 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$