Simplify (2+ square root of 8)/( square root of 6)

$\frac{2 + \sqrt{8}}{\sqrt{6}}$

Step 1

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $2 + 2 \sqrt{2}$ and $6$ .

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

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

$\frac{1 \sqrt{6} + \sqrt{2} \sqrt{6}}{3}$

Multiply $\sqrt{6}$ by $1$ .

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

Combine using the product rule for radicals.

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

Step 4

Simplify each term.

Multiply $2$ by $6$ .

$\frac{\sqrt{6} + \sqrt{12}}{3}$

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.97119711 \dots$

Simplify (2+ square root of 8)/( square root of 6)

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