Evaluate (3+ square root of 18)/(4 square root of 12)

$\frac{3 + \sqrt{18}}{4 \sqrt{12}}$

Step 1

Simplify the numerator.

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

Factor $9$ out of $18$ .

$\frac{3 + \sqrt{9 (2)}}{4 \sqrt{12}}$

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

$\frac{3 + \sqrt{3^{2} \cdot 2}}{4 \sqrt{12}}$

Pull terms out from under the radical.

$\frac{3 + 3 \sqrt{2}}{4 \sqrt{12}}$

Step 2

Simplify the denominator.

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

Factor $4$ out of $12$ .

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

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

$\frac{3 + 3 \sqrt{2}}{4 \sqrt{2^{2} \cdot 3}}$

Pull terms out from under the radical.

$\frac{3 + 3 \sqrt{2}}{4 \cdot 2 \sqrt{3}}$

Multiply $4$ by $2$ .

$\frac{3 + 3 \sqrt{2}}{8 \sqrt{3}}$

Step 3

Multiply $\frac{3 + 3 \sqrt{2}}{8 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{3 + 3 \sqrt{2}}{8 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{3 + 3 \sqrt{2}}{8 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 \sqrt{3} \sqrt{3}}$

Move $\sqrt{3}$ .

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 (\sqrt{3} \sqrt{3})}$

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

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 ({\sqrt{3}}^{1} \sqrt{3})}$

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

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

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

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 {\sqrt{3}}^{2}}$

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

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

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 {(3^{\frac{1}{2}})}^{2}}$

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

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 \cdot 3^{1}}$

Evaluate the exponent.

$\frac{(3 + 3 \sqrt{2}) \sqrt{3}}{8 \cdot 3}$

Step 5

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

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

$\frac{3 ((1 + \sqrt{2}) \sqrt{3})}{8 \cdot 3}$

Cancel the common factors.

Factor $3$ out of $8 \cdot 3$ .

$\frac{3 ((1 + \sqrt{2}) \sqrt{3})}{3 \cdot 8}$

Cancel the common factor.

$\frac{3 ((1 + \sqrt{2}) \sqrt{3})}{3 \cdot 8}$

Rewrite the expression.

$\frac{(1 + \sqrt{2}) \sqrt{3}}{8}$

Step 6

Apply the distributive property.

$\frac{1 \sqrt{3} + \sqrt{2} \sqrt{3}}{8}$

Step 7

Multiply $\sqrt{3}$ by $1$ .

$\frac{\sqrt{3} + \sqrt{2} \sqrt{3}}{8}$

Step 8

Combine using the product rule for radicals.

$\frac{\sqrt{3} + \sqrt{2 \cdot 3}}{8}$

Step 9

Multiply $2$ by $3$ .

$\frac{\sqrt{3} + \sqrt{6}}{8}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3} + \sqrt{6}}{8}$

Decimal Form:

$0.52269256 \dots$

Evaluate (3+ square root of 18)/(4 square root of 12)

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