Solve 3sqrt{22/3}*(-1/3sqrt{-5})div2/2sqrt{2/5}

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

Evaluate (complex solution)

$$-\frac{4\sqrt{3}i}{3}\approx -0-2.309401077i$$

Solution Steps

Divide 2 by 2 to get 1.

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

Multiply 2 and 3 to get 6.

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

Add 6 and 2 to get 8.

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

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

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

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}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3 and 3.

$$\frac{-\frac{2\sqrt{6}}{3}\sqrt{-5}}{1}\sqrt{\frac{2}{5}}$$

Factor -5=5\left(-1\right). Rewrite the square root of the product \sqrt{5\left(-1\right)} as the product of square roots \sqrt{5}\sqrt{-1}. By definition, the square root of -1 is i.

$$\frac{-\frac{2\sqrt{6}}{3}\sqrt{5}i}{1}\sqrt{\frac{2}{5}}$$

Multiply -1 and i to get -i.

$$\frac{-i\times \frac{2\sqrt{6}}{3}\sqrt{5}}{1}\sqrt{\frac{2}{5}}$$

Express \frac{2\sqrt{6}}{3}\sqrt{5} as a single fraction.

$$\frac{-i\times \frac{2\sqrt{6}\sqrt{5}}{3}}{1}\sqrt{\frac{2}{5}}$$

Anything divided by one gives itself.

$$-i\times \frac{2\sqrt{6}\sqrt{5}}{3}\sqrt{\frac{2}{5}}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$-i\times \frac{2\sqrt{30}}{3}\sqrt{\frac{2}{5}}$$

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

$$-i\times \frac{2\sqrt{30}}{3}\times \frac{\sqrt{2}}{\sqrt{5}}$$

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

$$-i\times \frac{2\sqrt{30}}{3}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$-i\times \frac{2\sqrt{30}}{3}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

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

$$-i\times \frac{2\sqrt{30}}{3}\times \frac{\sqrt{10}}{5}$$

Multiply \frac{2\sqrt{30}}{3} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$-i\times \frac{2\sqrt{30}\sqrt{10}}{3\times 5}$$

Factor 30=10\times 3. Rewrite the square root of the product \sqrt{10\times 3} as the product of square roots \sqrt{10}\sqrt{3}.

$$-i\times \frac{2\sqrt{10}\sqrt{3}\sqrt{10}}{3\times 5}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$-i\times \frac{2\times 10\sqrt{3}}{3\times 5}$$

Multiply 2 and 10 to get 20.

$$-i\times \frac{20\sqrt{3}}{3\times 5}$$

Multiply 3 and 5 to get 15.

$$-i\times \frac{20\sqrt{3}}{15}$$

Divide 20\sqrt{3} by 15 to get \frac{4}{3}\sqrt{3}.

$$-i\times \frac{4}{3}\sqrt{3}$$

Multiply -i and \frac{4}{3} to get -\frac{4}{3}i.

$$-\frac{4}{3}i\sqrt{3}$$

Real Part (complex solution)

$$0$$

Evaluate

$$\text{Indeterminate}$$