Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 \sqrt{6} - 5) (\sqrt{6} - 2)}{{\sqrt{6}}^{2} + \sqrt{6} \cdot - 2 + 2 \sqrt{6} - 4}$

Step 2.3

Simplify.

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

Step 3

Expand $(3 \sqrt{6} - 5) (\sqrt{6} - 2)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

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

Step 4.1.2.4.2

Rewrite the expression.

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

Step 4.1.2.5

Evaluate the exponent.

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

Step 4.1.3

Multiply $3$ by $6$ .

$\frac{18 + 3 \sqrt{6} \cdot - 2 - 5 \sqrt{6} - 5 \cdot - 2}{2}$

Step 4.1.4

Multiply $- 2$ by $3$ .

$\frac{18 - 6 \sqrt{6} - 5 \sqrt{6} - 5 \cdot - 2}{2}$

Step 4.1.5

Multiply $- 5$ by $- 2$ .

$\frac{18 - 6 \sqrt{6} - 5 \sqrt{6} + 10}{2}$

Step 4.2

Add $18$ and $10$ .

$\frac{28 - 6 \sqrt{6} - 5 \sqrt{6}}{2}$

Step 4.3

Subtract $5 \sqrt{6}$ from $- 6 \sqrt{6}$ .

$\frac{28 - 11 \sqrt{6}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{28 - 11 \sqrt{6}}{2}$

Decimal Form:

$0.52780641 \dots$