Solve 9sqrt{5}div3sqrt{1/125}*(-3/2sqrt{22/3})

$$9 \sqrt { 5 } \div 3 \sqrt { \frac { 1 } { 125 } } \times ( - \frac { 3 } { 2 } \sqrt { 2 \frac { 2 } { 3 } } )$$

Evaluate

$$-\frac{3\sqrt{6}}{5}\approx -1.469693846$$

Solution Steps

Divide 9\sqrt{5} by 3 to get 3\sqrt{5}.

$$3\sqrt{5}\sqrt{\frac{1}{125}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{125}} as the division of square roots \frac{\sqrt{1}}{\sqrt{125}}.

$$3\sqrt{5}\times \frac{\sqrt{1}}{\sqrt{125}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{5}\times \frac{1}{\sqrt{125}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

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

Rationalize the denominator of \frac{1}{5\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

Multiply 5 and 5 to get 25.

$$3\sqrt{5}\times \frac{\sqrt{5}}{25}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}$$

Express 3\left(-\frac{3}{2}\right) as a single fraction.

$$\frac{3\left(-3\right)}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 3 and -3 to get -9.

$$\frac{-9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\sqrt{\frac{2\times 3+2}{3}}$$

Fraction \frac{-9}{2} can be rewritten as -\frac{9}{2} by extracting the negative sign.

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 2 and 3 to get 6.

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\sqrt{\frac{6+2}{3}}$$

Add 6 and 2 to get 8.

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\sqrt{\frac{8}{3}}$$

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

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\times \frac{\sqrt{8}}{\sqrt{3}}$$

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}.

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\times \frac{2\sqrt{2}}{\sqrt{3}}$$

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

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\times \frac{2\sqrt{2}\sqrt{3}}{3}$$

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

$$-\frac{9}{2}\sqrt{5}\times \frac{\sqrt{5}}{25}\times \frac{2\sqrt{6}}{3}$$

Multiply -\frac{9}{2} times \frac{\sqrt{5}}{25} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-9\sqrt{5}}{2\times 25}\sqrt{5}\times \frac{2\sqrt{6}}{3}$$

Multiply \frac{-9\sqrt{5}}{2\times 25} times \frac{2\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-9\sqrt{5}\times 2\sqrt{6}}{2\times 25\times 3}\sqrt{5}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{-3\sqrt{5}\sqrt{6}}{25}\sqrt{5}$$

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

$$\frac{-3\sqrt{30}}{25}\sqrt{5}$$

Express \frac{-3\sqrt{30}}{25}\sqrt{5} as a single fraction.

$$\frac{-3\sqrt{30}\sqrt{5}}{25}$$

Factor 30=5\times 6. Rewrite the square root of the product \sqrt{5\times 6} as the product of square roots \sqrt{5}\sqrt{6}.

$$\frac{-3\sqrt{5}\sqrt{6}\sqrt{5}}{25}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{-3\times 5\sqrt{6}}{25}$$

Multiply -3 and 5 to get -15.

$$\frac{-15\sqrt{6}}{25}$$

Divide -15\sqrt{6} by 25 to get -\frac{3}{5}\sqrt{6}.

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