Algebra Examples

$\frac{\sqrt{5} - \sqrt{6}}{\sqrt{18}}$

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{5} - \sqrt{6}}{\sqrt{9 (2)}}$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{\sqrt{5} - \sqrt{6}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{5} - \sqrt{6}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.2

Move $\sqrt{2}$ .

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

Step 3.3

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

$\frac{(\sqrt{5} - \sqrt{6}) \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.4

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

$\frac{(\sqrt{5} - \sqrt{6}) \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.5

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

$\frac{(\sqrt{5} - \sqrt{6}) \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

$\frac{(\sqrt{5} - \sqrt{6}) \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 3.7.2

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

$\frac{(\sqrt{5} - \sqrt{6}) \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

$\frac{(\sqrt{5} - \sqrt{6}) \sqrt{2}}{3 \cdot 2^{1}}$

Step 3.7.5

Evaluate the exponent.

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

Step 4

Multiply $3$ by $2$ .

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

Step 5

Apply the distributive property.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

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

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $2$ by $6$ .

$\frac{\sqrt{5 \cdot 2} - \sqrt{12}}{6}$

Step 8

Simplify each term.

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Step 8.1

Multiply $5$ by $2$ .

$\frac{\sqrt{10} - \sqrt{12}}{6}$

Step 8.2

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

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Step 8.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{10} - \sqrt{4 (3)}}{6}$

Step 8.2.2

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

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

Step 8.3

Pull terms out from under the radical.

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

Step 8.4

Multiply $2$ by $- 1$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.05030399 \dots$