Solve sqrt{8/3}-sqrt{3/8}

$$\sqrt { \frac { 8 } { 3 } } - \sqrt { \frac { 3 } { 8 } }$$

Evaluate

$$\frac{5\sqrt{6}}{12}\approx 1.020620726$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\sqrt{8}}{\sqrt{3}}-\sqrt{\frac{3}{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{\sqrt{3}}-\sqrt{\frac{3}{8}}$$

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

$$\frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{\frac{3}{8}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{2}\sqrt{3}}{3}-\sqrt{\frac{3}{8}}$$

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

$$\frac{2\sqrt{6}}{3}-\sqrt{\frac{3}{8}}$$

Rewrite the square root of the division \sqrt{\frac{3}{8}} as the division of square roots \frac{\sqrt{3}}{\sqrt{8}}.

$$\frac{2\sqrt{6}}{3}-\frac{\sqrt{3}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply 2 and 2 to get 4.

$$\frac{2\sqrt{6}}{3}-\frac{\sqrt{6}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{2\sqrt{6}}{3} times \frac{4}{4}. Multiply \frac{\sqrt{6}}{4} times \frac{3}{3}.

$$\frac{4\times 2\sqrt{6}}{12}-\frac{3\sqrt{6}}{12}$$

Since \frac{4\times 2\sqrt{6}}{12} and \frac{3\sqrt{6}}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\times 2\sqrt{6}-3\sqrt{6}}{12}$$

Do the multiplications in 4\times 2\sqrt{6}-3\sqrt{6}.

$$\frac{8\sqrt{6}-3\sqrt{6}}{12}$$

Do the calculations in 8\sqrt{6}-3\sqrt{6}.

$$\frac{5\sqrt{6}}{12}$$