Solve sqrt{(3/2-5/3)(-1/3)}

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

Evaluate

$$\frac{\sqrt{2}}{6}\approx 0.23570226$$

Solution Steps

Least common multiple of 2 and 3 is 6. Convert \frac{3}{2} and \frac{5}{3} to fractions with denominator 6.

$$\sqrt{\left(\frac{9}{6}-\frac{10}{6}\right)\left(-\frac{1}{3}\right)}$$

Since \frac{9}{6} and \frac{10}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{9-10}{6}\left(-\frac{1}{3}\right)}$$

Subtract 10 from 9 to get -1.

$$\sqrt{-\frac{1}{6}\left(-\frac{1}{3}\right)}$$

Multiply -\frac{1}{6} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{-\left(-1\right)}{6\times 3}}$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{6\times 3}.

$$\sqrt{\frac{1}{18}}$$

Rewrite the square root of the division \sqrt{\frac{1}{18}} as the division of square roots \frac{\sqrt{1}}{\sqrt{18}}.

$$\frac{\sqrt{1}}{\sqrt{18}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{18}}$$

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}.

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

Rationalize the denominator of \frac{1}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 3 and 2 to get 6.

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