Solve 2sqrt{3}+sqrt{27}-sqrt{1/2}

$$2 \sqrt { 3 } + \sqrt { 27 } - \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$-\frac{\sqrt{2}}{2}+5\sqrt{3}\approx 7.953147257$$

Short Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

Combine 2\sqrt{3} and 3\sqrt{3} to get 5\sqrt{3}.

$$5\sqrt{3}-\sqrt{\frac{1}{2}}$$

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

$$5\sqrt{3}-\frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$5\sqrt{3}-\frac{1}{\sqrt{2}}$$

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

$$5\sqrt{3}-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$5\sqrt{3}-\frac{\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5\sqrt{3} times \frac{2}{2}.

$$\frac{2\times 5\sqrt{3}}{2}-\frac{\sqrt{2}}{2}$$

Since \frac{2\times 5\sqrt{3}}{2} and \frac{\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 5\sqrt{3}-\sqrt{2}}{2}$$

Do the multiplications in 2\times 5\sqrt{3}-\sqrt{2}.

$$\frac{10\sqrt{3}-\sqrt{2}}{2}$$