Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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Step 2.1.5.1

Combine using the product rule for radicals.

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

Step 2.1.5.2

Multiply $2$ by $3$ .

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

Step 2.1.6

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

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

Step 2.1.7

Combine using the product rule for radicals.

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

Step 2.1.8

Multiply $3$ by $2$ .

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

Step 2.1.9

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

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Step 2.1.9.1

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

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

Step 2.1.9.2

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

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

Step 2.1.9.3

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

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

Step 2.1.9.4

Add $1$ and $1$ .

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

Step 2.1.10

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

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Cancel the common factor of $2$ .

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Step 2.1.10.4.1

Cancel the common factor.

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

Step 2.1.10.4.2

Rewrite the expression.

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

Step 2.1.10.5

Evaluate the exponent.

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

Step 2.1.11

Multiply $- 1$ by $3$ .

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

Step 2.1.12

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

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $3$ from $2$ .

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

Step 2.2.2

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

$- 1 + 0 + \sqrt{2} + \sqrt{3}$

Step 2.2.3

Add $- 1$ and $0$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.14626436 \dots$