Algebra Examples

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

Step 1

Expand $(\sqrt{3} - \sqrt{2} + \sqrt{6}) (\sqrt{6} - \sqrt{3})$ by multiplying each term in the first expression by each term in the second expression.

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

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.

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

Step 2.1.2

Multiply $3$ by $6$ .

$\sqrt{18} + \sqrt{3} (- \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{3}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{3})$

Step 2.1.3

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.3.1

Factor $9$ out of $18$ .

$\sqrt{9 (2)} + \sqrt{3} (- \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{3}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{3})$

Step 2.1.3.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.4

Pull terms out from under the radical.

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Add $1$ and $1$ .

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

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

Step 2.1.7

Multiply $- 1$ by $3$ .

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

Step 2.1.8

Multiply $- \sqrt{2} \sqrt{6}$ .

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Step 2.1.8.1

Combine using the product rule for radicals.

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

Step 2.1.8.2

Multiply $6$ by $2$ .

$3 \sqrt{2} - 3 - \sqrt{12} - \sqrt{2} (- \sqrt{3}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{3})$

Step 2.1.9

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.9.1

Factor $4$ out of $12$ .

$3 \sqrt{2} - 3 - \sqrt{4 (3)} - \sqrt{2} (- \sqrt{3}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{3})$

Step 2.1.9.2

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

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

Step 2.1.10

Pull terms out from under the radical.

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

Step 2.1.11

Multiply $2$ by $- 1$ .

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

Step 2.1.12

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

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Step 2.1.12.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.12.2

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

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

Step 2.1.12.3

Combine using the product rule for radicals.

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

Step 2.1.12.4

Multiply $2$ by $3$ .

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

Step 2.1.13

Combine using the product rule for radicals.

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

Step 2.1.14

Multiply $6$ by $6$ .

$3 \sqrt{2} - 3 - 2 \sqrt{3} + \sqrt{6} + \sqrt{36} + \sqrt{6} (- \sqrt{3})$

Step 2.1.15

Rewrite $36$ as $6^{2}$ .

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

Step 2.1.16

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

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

Step 2.1.17

Multiply $\sqrt{6} (- \sqrt{3})$ .

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Step 2.1.17.1

Combine using the product rule for radicals.

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

Step 2.1.17.2

Multiply $6$ by $3$ .

$3 \sqrt{2} - 3 - 2 \sqrt{3} + \sqrt{6} + 6 - \sqrt{18}$

Step 2.1.18

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.18.1

Factor $9$ out of $18$ .

$3 \sqrt{2} - 3 - 2 \sqrt{3} + \sqrt{6} + 6 - \sqrt{9 (2)}$

Step 2.1.18.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.19

Pull terms out from under the radical.

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

Step 2.1.20

Multiply $3$ by $- 1$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $3 \sqrt{2}$ from $3 \sqrt{2}$ .

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

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $3$ from $0$ .

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

Step 2.2.2.2

Add $- 3$ and $6$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.98538812 \dots$