Solve frac{sqrt{2}}{sqrt{12}+4sqrt{3}}

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

Evaluate

$$\frac{\sqrt{6}}{18}\approx 0.136082763$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

Combine 2\sqrt{3} and 4\sqrt{3} to get 6\sqrt{3}.

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

Rationalize the denominator of \frac{\sqrt{2}}{6\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

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

Multiply 6 and 3 to get 18.

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