Simplify (3 square root of 2+ square root of 6)/( square root of 12)

$\frac{3 \sqrt{2} + \sqrt{6}}{\sqrt{12}}$

Step 1

Simplify the denominator.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Move $\sqrt{3}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 4

Multiply $2$ by $3$ .

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

Step 5

Apply the distributive property.

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

Step 6

Multiply $3 \sqrt{2} \sqrt{3}$ .

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Step 7

Combine using the product rule for radicals.

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

Step 8

Simplify each term.

Multiply $6$ by $3$ .

$\frac{3 \sqrt{6} + \sqrt{18}}{6}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

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

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Step 9

Cancel the common factor of $3 \sqrt{6} + 3 \sqrt{2}$ and $6$ .

Factor $3$ out of $3 \sqrt{6}$ .

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

Factor $3$ out of $3 \sqrt{2}$ .

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

Factor $3$ out of $3 (\sqrt{6}) + 3 (\sqrt{2})$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.93185165 \dots$

Simplify (3 square root of 2+ square root of 6)/( square root of 12)

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