Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Apply the distributive property.

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

Step 4

Combine using the product rule for radicals.

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

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply $6$ by $2$ .

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

Step 6

Simplify each term.

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Step 6.1

Multiply $5$ by $6$ .

$\frac{\sqrt{30} + 2 \sqrt{12}}{6}$

Step 6.2

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

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Step 6.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{30} + 2 \sqrt{4 (3)}}{6}$

Step 6.2.2

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

$\frac{\sqrt{30} + 2 \sqrt{2^{2} \cdot 3}}{6}$

Step 6.3

Pull terms out from under the radical.

$\frac{\sqrt{30} + 2 (2 \sqrt{3})}{6}$

Step 6.4

Multiply $2$ by $2$ .

$\frac{\sqrt{30} + 4 \sqrt{3}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30} + 4 \sqrt{3}}{6}$

Decimal Form:

$2.06757146 \dots$