Solve 2sqrt{18}-3sqrt{2}-3sqrt{1/2}

$$2 \sqrt { 18 } - 3 \sqrt { 2 } - 3 \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$\frac{3\sqrt{2}}{2}\approx 2.121320344$$

Short Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

Multiply 2 and 3 to get 6.

$$6\sqrt{2}-3\sqrt{2}-3\sqrt{\frac{1}{2}}$$

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

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

Express -3\times \frac{\sqrt{2}}{2} as a single fraction.

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

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

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

Since \frac{2\times 3\sqrt{2}}{2} and \frac{-3\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

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

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

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

Do the calculations in 6\sqrt{2}-3\sqrt{2}.

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