Algebra Examples

$\frac{3 + \sqrt{6}}{5 - \sqrt{24}}$

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $24$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(3 + \sqrt{6}) (5 + 2 \sqrt{6})}{25 + 10 \sqrt{6} - 10 \sqrt{6} - 4 {\sqrt{6}}^{2}}$

Step 3.3

Simplify.

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

Step 3.4

Divide $(3 + \sqrt{6}) (5 + 2 \sqrt{6})$ by $1$ .

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $3$ by $5$ .

$15 + 3 (2 \sqrt{6}) + \sqrt{6} \cdot 5 + \sqrt{6} (2 \sqrt{6})$

Step 5.1.2

Multiply $2$ by $3$ .

$15 + 6 \sqrt{6} + \sqrt{6} \cdot 5 + \sqrt{6} (2 \sqrt{6})$

Step 5.1.3

Move $5$ to the left of $\sqrt{6}$ .

$15 + 6 \sqrt{6} + 5 \cdot \sqrt{6} + \sqrt{6} (2 \sqrt{6})$

Step 5.1.4

Multiply $\sqrt{6} (2 \sqrt{6})$ .

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Step 5.1.4.1

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

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 ({\sqrt{6}}^{1} \sqrt{6})$

Step 5.1.4.2

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

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})$

Step 5.1.4.3

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

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 {\sqrt{6}}^{1 + 1}$

Step 5.1.4.4

Add $1$ and $1$ .

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 {\sqrt{6}}^{2}$

Step 5.1.5

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

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Step 5.1.5.1

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

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 {(6^{\frac{1}{2}})}^{2}$

Step 5.1.5.2

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

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 \cdot 6^{\frac{1}{2} \cdot 2}$

Step 5.1.5.3

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

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 \cdot 6^{\frac{2}{2}}$

Step 5.1.5.4

Cancel the common factor of $2$ .

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Step 5.1.5.4.1

Cancel the common factor.

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 \cdot 6^{\frac{2}{2}}$

Step 5.1.5.4.2

Rewrite the expression.

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 \cdot 6^{1}$

Step 5.1.5.5

Evaluate the exponent.

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 2 \cdot 6$

Step 5.1.6

Multiply $2$ by $6$ .

$15 + 6 \sqrt{6} + 5 \sqrt{6} + 12$

Step 5.2

Add $15$ and $12$ .

$27 + 6 \sqrt{6} + 5 \sqrt{6}$

Step 5.3

Add $6 \sqrt{6}$ and $5 \sqrt{6}$ .

$27 + 11 \sqrt{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$27 + 11 \sqrt{6}$

Decimal Form:

$53.94438717 \dots$