Algebra Examples

$\frac{\sqrt{10} - 3 \sqrt{6}}{\sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{10} - 3 \sqrt{6}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{10} - 3 \sqrt{6}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{10} - 3 \sqrt{6}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4

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

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{2^{1}}$

Step 2.6.5

Evaluate the exponent.

$\frac{(\sqrt{10} - 3 \sqrt{6}) \sqrt{2}}{2}$

Step 3

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{2} - 3 \sqrt{6} \sqrt{2}}{2}$

Step 4

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 2} - 3 \sqrt{6} \sqrt{2}}{2}$

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 2} - 3 \sqrt{2 \cdot 6}}{2}$

Step 5.2

Multiply $2$ by $6$ .

$\frac{\sqrt{10 \cdot 2} - 3 \sqrt{12}}{2}$

Step 6

Simplify each term.

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Step 6.1

Multiply $10$ by $2$ .

$\frac{\sqrt{20} - 3 \sqrt{12}}{2}$

Step 6.2

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.2.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)} - 3 \sqrt{12}}{2}$

Step 6.2.2

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

$\frac{\sqrt{2^{2} \cdot 5} - 3 \sqrt{12}}{2}$

Step 6.3

Pull terms out from under the radical.

$\frac{2 \sqrt{5} - 3 \sqrt{12}}{2}$

Step 6.4

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

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Step 6.4.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{5} - 3 \sqrt{4 (3)}}{2}$

Step 6.4.2

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

$\frac{2 \sqrt{5} - 3 \sqrt{2^{2} \cdot 3}}{2}$

Step 6.5

Pull terms out from under the radical.

$\frac{2 \sqrt{5} - 3 (2 \sqrt{3})}{2}$

Step 6.6

Multiply $2$ by $- 3$ .

$\frac{2 \sqrt{5} - 6 \sqrt{3}}{2}$

Step 7

Cancel the common factor of $2 \sqrt{5} - 6 \sqrt{3}$ and $2$ .

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Step 7.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 (\sqrt{5}) - 6 \sqrt{3}}{2}$

Step 7.2

Factor $2$ out of $- 6 \sqrt{3}$ .

$\frac{2 (\sqrt{5}) + 2 (- 3 \sqrt{3})}{2}$

Step 7.3

Factor $2$ out of $2 (\sqrt{5}) + 2 (- 3 \sqrt{3})$ .

$\frac{2 (\sqrt{5} - 3 \sqrt{3})}{2}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $2$ out of $2$ .

$\frac{2 (\sqrt{5} - 3 \sqrt{3})}{2 (1)}$

Step 7.4.2

Cancel the common factor.

$\frac{2 (\sqrt{5} - 3 \sqrt{3})}{2 \cdot 1}$

Step 7.4.3

Rewrite the expression.

$\frac{\sqrt{5} - 3 \sqrt{3}}{1}$

Step 7.4.4

Divide $\sqrt{5} - 3 \sqrt{3}$ by $1$ .

$\sqrt{5} - 3 \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{5} - 3 \sqrt{3}$

Decimal Form:

$- 2.96008444 \dots$