Solve 3sqrt{22/3}+1/2sqrt{2/5}*(-1/8sqrt{15})

$$3 \sqrt { 2 \frac { 2 } { 3 } } + \frac { 1 } { 2 } \sqrt { \frac { 2 } { 5 } } \times ( - \frac { 1 } { 8 } \sqrt { 15 } )$$

Evaluate

$$\frac{31\sqrt{6}}{16}\approx 4.745886377$$

Short Solution Steps

Multiply 2 and 3 to get 6.

$$3\sqrt{\frac{6+2}{3}}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

Add 6 and 2 to get 8.

$$3\sqrt{\frac{8}{3}}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

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

$$3\times \frac{\sqrt{8}}{\sqrt{3}}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

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}.

$$3\times \frac{2\sqrt{2}}{\sqrt{3}}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

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

$$3\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

The square of \sqrt{3} is 3.

$$3\times \frac{2\sqrt{2}\sqrt{3}}{3}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

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

$$3\times \frac{2\sqrt{6}}{3}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

Cancel out 3 and 3.

$$2\sqrt{6}+\frac{1}{2}\sqrt{\frac{2}{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

Rewrite the square root of the division \sqrt{\frac{2}{5}} as the division of square roots \frac{\sqrt{2}}{\sqrt{5}}.

$$2\sqrt{6}+\frac{1}{2}\times \frac{\sqrt{2}}{\sqrt{5}}\left(-\frac{1}{8}\right)\sqrt{15}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$2\sqrt{6}+\frac{1}{2}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\left(-\frac{1}{8}\right)\sqrt{15}$$

The square of \sqrt{5} is 5.

$$2\sqrt{6}+\frac{1}{2}\times \frac{\sqrt{2}\sqrt{5}}{5}\left(-\frac{1}{8}\right)\sqrt{15}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$2\sqrt{6}+\frac{1}{2}\times \frac{\sqrt{10}}{5}\left(-\frac{1}{8}\right)\sqrt{15}$$

Multiply \frac{1}{2} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$2\sqrt{6}+\frac{1\left(-1\right)}{2\times 8}\times \frac{\sqrt{10}}{5}\sqrt{15}$$

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 8}.

$$2\sqrt{6}+\frac{-1}{16}\times \frac{\sqrt{10}}{5}\sqrt{15}$$

Fraction \frac{-1}{16} can be rewritten as -\frac{1}{16} by extracting the negative sign.

$$2\sqrt{6}-\frac{1}{16}\times \frac{\sqrt{10}}{5}\sqrt{15}$$

Multiply -\frac{1}{16} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$2\sqrt{6}+\frac{-\sqrt{10}}{16\times 5}\sqrt{15}$$

Express \frac{-\sqrt{10}}{16\times 5}\sqrt{15} as a single fraction.

$$2\sqrt{6}+\frac{-\sqrt{10}\sqrt{15}}{16\times 5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{6} times \frac{16\times 5}{16\times 5}.

$$\frac{2\sqrt{6}\times 16\times 5}{16\times 5}+\frac{-\sqrt{10}\sqrt{15}}{16\times 5}$$

Since \frac{2\sqrt{6}\times 16\times 5}{16\times 5} and \frac{-\sqrt{10}\sqrt{15}}{16\times 5} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{6}\times 16\times 5-\sqrt{10}\sqrt{15}}{16\times 5}$$

Do the multiplications in 2\sqrt{6}\times 16\times 5-\sqrt{10}\sqrt{15}.

$$\frac{160\sqrt{6}-5\sqrt{6}}{16\times 5}$$

Do the calculations in 160\sqrt{6}-5\sqrt{6}.

$$\frac{155\sqrt{6}}{16\times 5}$$

Cancel out 5 in both numerator and denominator.

$$\frac{31\sqrt{6}}{16}$$