Basic Math Examples

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

Step 1

Rewrite ${(\sqrt{3} - \sqrt{6})}^{2}$ as $(\sqrt{3} - \sqrt{6}) (\sqrt{3} - \sqrt{6})$ .

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

Step 2

Expand $(\sqrt{3} - \sqrt{6}) (\sqrt{3} - \sqrt{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine using the product rule for radicals.

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

Step 3.1.2

Multiply $3$ by $3$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

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

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Step 3.1.5.1

Combine using the product rule for radicals.

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

Step 3.1.5.2

Multiply $3$ by $6$ .

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

Step 3.1.6

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

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Step 3.1.6.1

Factor $9$ out of $18$ .

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

Step 3.1.6.2

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

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

Step 3.1.7

Pull terms out from under the radical.

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

Step 3.1.8

Multiply $3$ by $- 1$ .

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

Step 3.1.9

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

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Step 3.1.9.1

Combine using the product rule for radicals.

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

Step 3.1.9.2

Multiply $3$ by $6$ .

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

Step 3.1.10

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

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Step 3.1.10.1

Factor $9$ out of $18$ .

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

Step 3.1.10.2

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

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

Step 3.1.11

Pull terms out from under the radical.

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

Step 3.1.12

Multiply $3$ by $- 1$ .

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

Step 3.1.13

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

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Step 3.1.13.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.13.2

Multiply $\sqrt{6}$ by $1$ .

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

Step 3.1.13.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 3.1.13.4

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 3.1.13.5

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

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

Step 3.1.13.6

Add $1$ and $1$ .

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

Step 3.1.14

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.1.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 3.1.14.2

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

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

Step 3.1.14.3

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

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

Step 3.1.14.4

Cancel the common factor of $2$ .

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Step 3.1.14.4.1

Cancel the common factor.

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

Step 3.1.14.4.2

Rewrite the expression.

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

Step 3.1.14.5

Evaluate the exponent.

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

Step 3.2

Add $3$ and $6$ .

$9 - 3 \sqrt{2} - 3 \sqrt{2}$

Step 3.3

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

$9 - 6 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$9 - 6 \sqrt{2}$

Decimal Form:

$0.51471862 \dots$