Solve {l}{sqrt[2]{18}+6sqrt{1/2}-5sqrt{6}}{+sqrt{3}}

$$\left. \begin{array} { l } { \sqrt[ 2 ] { 18 } + 6 \sqrt { \frac { 1 } { 2 } } - 5 \sqrt { 6 } } \\ { + \sqrt { 3 } } \end{array} \right.$$

Evaluate

$$\sqrt{3}+6\sqrt{2}-5\sqrt{6}\approx -2.030116532$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$\sqrt[2]{18}+6\times \frac{\sqrt{1}}{\sqrt{2}}-5\sqrt{6}+\sqrt{3}$$

Calculate the square root of 1 and get 1.

$$\sqrt[2]{18}+6\times \frac{1}{\sqrt{2}}-5\sqrt{6}+\sqrt{3}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\sqrt[2]{18}+6\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-5\sqrt{6}+\sqrt{3}$$

The square of \sqrt{2} is 2.

$$\sqrt[2]{18}+6\times \frac{\sqrt{2}}{2}-5\sqrt{6}+\sqrt{3}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$\sqrt[2]{18}+3\sqrt{2}-5\sqrt{6}+\sqrt{3}$$