Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{6} (3 \sqrt{6} - 2)}{9 {\sqrt{6}}^{2} - 6 \sqrt{6} + 6 \sqrt{6} - 4}$

Step 4

Simplify.

$\frac{\sqrt{6} (3 \sqrt{6} - 2)}{50}$

Step 5

Apply the distributive property.

$\frac{\sqrt{6} (3 \sqrt{6}) + \sqrt{6} \cdot - 2}{50}$

Step 6

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

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Step 6.1

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

$\frac{3 ({\sqrt{6}}^{1} \sqrt{6}) + \sqrt{6} \cdot - 2}{50}$

Step 6.2

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

$\frac{3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + \sqrt{6} \cdot - 2}{50}$

Step 6.3

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

$\frac{3 {\sqrt{6}}^{1 + 1} + \sqrt{6} \cdot - 2}{50}$

Step 6.4

Add $1$ and $1$ .

$\frac{3 {\sqrt{6}}^{2} + \sqrt{6} \cdot - 2}{50}$

Step 7

Move $- 2$ to the left of $\sqrt{6}$ .

$\frac{3 {\sqrt{6}}^{2} - 2 \cdot \sqrt{6}}{50}$

Step 8

Simplify each term.

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Step 8.1

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

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Step 8.1.1

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

$\frac{3 {(6^{\frac{1}{2}})}^{2} - 2 \cdot \sqrt{6}}{50}$

Step 8.1.2

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

$\frac{3 \cdot 6^{\frac{1}{2} \cdot 2} - 2 \cdot \sqrt{6}}{50}$

Step 8.1.3

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

$\frac{3 \cdot 6^{\frac{2}{2}} - 2 \cdot \sqrt{6}}{50}$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Cancel the common factor.

$\frac{3 \cdot 6^{\frac{2}{2}} - 2 \cdot \sqrt{6}}{50}$

Step 8.1.4.2

Rewrite the expression.

$\frac{3 \cdot 6^{1} - 2 \cdot \sqrt{6}}{50}$

Step 8.1.5

Evaluate the exponent.

$\frac{3 \cdot 6 - 2 \cdot \sqrt{6}}{50}$

Step 8.2

Multiply $3$ by $6$ .

$\frac{18 - 2 \sqrt{6}}{50}$

Step 9

Cancel the common factor of $18 - 2 \sqrt{6}$ and $50$ .

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Step 9.1

Factor $2$ out of $18$ .

$\frac{2 (9) - 2 \sqrt{6}}{50}$

Step 9.2

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

$\frac{2 (9) + 2 (- \sqrt{6})}{50}$

Step 9.3

Factor $2$ out of $2 (9) + 2 (- \sqrt{6})$ .

$\frac{2 (9 - \sqrt{6})}{50}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $2$ out of $50$ .

$\frac{2 (9 - \sqrt{6})}{2 \cdot 25}$

Step 9.4.2

Cancel the common factor.

$\frac{2 (9 - \sqrt{6})}{2 \cdot 25}$

Step 9.4.3

Rewrite the expression.

$\frac{9 - \sqrt{6}}{25}$

Step 10

Split the fraction $\frac{9 - \sqrt{6}}{25}$ into two fractions.

$\frac{9}{25} + \frac{- \sqrt{6}}{25}$

Step 11

Move the negative in front of the fraction.

$\frac{9}{25} - \frac{\sqrt{6}}{25}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{25} - \frac{\sqrt{6}}{25}$

Decimal Form:

$0.26202041 \dots$