Algebra Examples

$(\sqrt{3} - \sqrt{6} + \sqrt{12} - \sqrt{24}) \cdot \frac{\sqrt{6}}{2}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

$(\sqrt{3} - \sqrt{6} + \sqrt{4 (3)} - \sqrt{24}) \cdot \frac{\sqrt{6}}{2}$

Step 1.1.2

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

$(\sqrt{3} - \sqrt{6} + \sqrt{2^{2} \cdot 3} - \sqrt{24}) \cdot \frac{\sqrt{6}}{2}$

Step 1.2

Pull terms out from under the radical.

$(\sqrt{3} - \sqrt{6} + | 2 | \sqrt{3} - \sqrt{24}) \cdot \frac{\sqrt{6}}{2}$

Step 1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$(\sqrt{3} - \sqrt{6} + 2 \sqrt{3} - \sqrt{24}) \cdot \frac{\sqrt{6}}{2}$

Step 1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 1.4.1

Factor $4$ out of $24$ .

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

Step 1.4.2

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

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

Step 1.5

Pull terms out from under the radical.

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

Step 1.6

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 1.7

Multiply $2$ by $- 1$ .

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

Step 2

Add $\sqrt{3}$ and $2 \sqrt{3}$ .

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

Step 3

Subtract $2 \sqrt{6}$ from $- \sqrt{6}$ .

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

Step 4

Apply the distributive property.

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

Step 5

Multiply $3 \sqrt{3} \frac{\sqrt{6}}{2}$ .

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Step 5.1

Combine $\frac{\sqrt{6}}{2}$ and $3$ .

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

Step 5.2

Combine $\frac{\sqrt{6} \cdot 3}{2}$ and $\sqrt{3}$ .

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

Step 5.3

Combine using the product rule for radicals.

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

Step 5.4

Multiply $6$ by $3$ .

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

Step 6

Multiply $- 3 \sqrt{6} \frac{\sqrt{6}}{2}$ .

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Step 6.1

Combine $\frac{\sqrt{6}}{2}$ and $- 3$ .

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

Step 6.2

Combine $\frac{\sqrt{6} \cdot - 3}{2}$ and $\sqrt{6}$ .

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

Step 6.3

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

$\frac{3 \sqrt{18}}{2} + \frac{- 3 ({\sqrt{6}}^{1} \sqrt{6})}{2}$

Step 6.4

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

$\frac{3 \sqrt{18}}{2} + \frac{- 3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}{2}$

Step 6.5

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

$\frac{3 \sqrt{18}}{2} + \frac{- 3 {\sqrt{6}}^{1 + 1}}{2}$

Step 6.6

Add $1$ and $1$ .

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

Step 7

Simplify each term.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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Step 7.1.1.1

Factor $9$ out of $18$ .

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

Step 7.1.1.2

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

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

Step 7.1.2

Pull terms out from under the radical.

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

Step 7.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 7.1.4

Multiply $3$ by $3$ .

$\frac{9 \sqrt{2}}{2} + \frac{- 3 {\sqrt{6}}^{2}}{2}$

Step 7.2

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

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Step 7.2.1

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

$\frac{9 \sqrt{2}}{2} + \frac{- 3 {(6^{\frac{1}{2}})}^{2}}{2}$

Step 7.2.2

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

$\frac{9 \sqrt{2}}{2} + \frac{- 3 \cdot 6^{\frac{1}{2} \cdot 2}}{2}$

Step 7.2.3

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

$\frac{9 \sqrt{2}}{2} + \frac{- 3 \cdot 6^{\frac{2}{2}}}{2}$

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

$\frac{9 \sqrt{2}}{2} + \frac{- 3 \cdot 6^{\frac{2}{2}}}{2}$

Step 7.2.4.2

Rewrite the expression.

$\frac{9 \sqrt{2}}{2} + \frac{- 3 \cdot 6^{1}}{2}$

Step 7.2.5

Evaluate the exponent.

$\frac{9 \sqrt{2}}{2} + \frac{- 3 \cdot 6}{2}$

Step 7.3

Multiply $- 3$ by $6$ .

$\frac{9 \sqrt{2}}{2} + \frac{- 18}{2}$

Step 7.4

Divide $- 18$ by $2$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 2.63603896 \dots$