Solve 14sqrt{3/2}-3sqrt{1/2})-(1/4sqrt{8}-3sqrt{2/3})

$$14 \sqrt { \frac { 3 } { 2 } } - 3 \sqrt { \frac { 1 } { 2 } } ) - ( \frac { 1 } { 4 } \sqrt { 8 } - 3 \sqrt { \frac { 2 } { 3 } } )$$

Evaluate

$$-\frac{3\sqrt{2}}{2}+7\sqrt{6}\approx 15.025107856$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$14\times \frac{\sqrt{3}}{\sqrt{2}}-3\sqrt{\frac{1}{2}}$$

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

$$14\times \frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-3\sqrt{\frac{1}{2}}$$

The square of \sqrt{2} is 2.

$$14\times \frac{\sqrt{3}\sqrt{2}}{2}-3\sqrt{\frac{1}{2}}$$

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

$$14\times \frac{\sqrt{6}}{2}-3\sqrt{\frac{1}{2}}$$

Cancel out 2, the greatest common factor in 14 and 2.

$$7\sqrt{6}-3\sqrt{\frac{1}{2}}$$

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

$$7\sqrt{6}-3\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$7\sqrt{6}-3\times \frac{1}{\sqrt{2}}$$

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

$$7\sqrt{6}-3\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$7\sqrt{6}-3\times \frac{\sqrt{2}}{2}$$

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

$$7\sqrt{6}+\frac{-3\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 7\sqrt{6} times \frac{2}{2}.

$$\frac{2\times 7\sqrt{6}}{2}+\frac{-3\sqrt{2}}{2}$$

Since \frac{2\times 7\sqrt{6}}{2} and \frac{-3\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\times 7\sqrt{6}-3\sqrt{2}}{2}$$

Do the multiplications in 2\times 7\sqrt{6}-3\sqrt{2}.

$$\frac{14\sqrt{6}-3\sqrt{2}}{2}$$