Solve 6sqrt{75}*(-3/2*sqrt{33/4})div(-9sqrt{20})

$$6 \sqrt { 75 } \times ( - \frac { 3 } { 2 } \times \sqrt { 3 \frac { 3 } { 4 } } ) \div ( - 9 \sqrt { 20 } )$$

Evaluate

$$\frac{15}{4}=3.75$$

Solution Steps

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{6\times 5\sqrt{3}\left(-\frac{3}{2}\right)\sqrt{\frac{3\times 4+3}{4}}}{-9\sqrt{20}}$$

Multiply 6 and 5 to get 30.

$$\frac{30\sqrt{3}\left(-\frac{3}{2}\right)\sqrt{\frac{3\times 4+3}{4}}}{-9\sqrt{20}}$$

Express 30\left(-\frac{3}{2}\right) as a single fraction.

$$\frac{\frac{30\left(-3\right)}{2}\sqrt{3}\sqrt{\frac{3\times 4+3}{4}}}{-9\sqrt{20}}$$

Multiply 30 and -3 to get -90.

$$\frac{\frac{-90}{2}\sqrt{3}\sqrt{\frac{3\times 4+3}{4}}}{-9\sqrt{20}}$$

Divide -90 by 2 to get -45.

$$\frac{-45\sqrt{3}\sqrt{\frac{3\times 4+3}{4}}}{-9\sqrt{20}}$$

Multiply 3 and 4 to get 12.

$$\frac{-45\sqrt{3}\sqrt{\frac{12+3}{4}}}{-9\sqrt{20}}$$

Add 12 and 3 to get 15.

$$\frac{-45\sqrt{3}\sqrt{\frac{15}{4}}}{-9\sqrt{20}}$$

Rewrite the square root of the division \sqrt{\frac{15}{4}} as the division of square roots \frac{\sqrt{15}}{\sqrt{4}}.

$$\frac{-45\sqrt{3}\times \frac{\sqrt{15}}{\sqrt{4}}}{-9\sqrt{20}}$$

Calculate the square root of 4 and get 2.

$$\frac{-45\sqrt{3}\times \frac{\sqrt{15}}{2}}{-9\sqrt{20}}$$

Express -45\times \frac{\sqrt{15}}{2} as a single fraction.

$$\frac{\frac{-45\sqrt{15}}{2}\sqrt{3}}{-9\sqrt{20}}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\frac{-45\sqrt{15}}{2}\sqrt{3}}{-9\times 2\sqrt{5}}$$

Multiply -9 and 2 to get -18.

$$\frac{\frac{-45\sqrt{15}}{2}\sqrt{3}}{-18\sqrt{5}}$$

Rationalize the denominator of \frac{\frac{-45\sqrt{15}}{2}\sqrt{3}}{-18\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{-45\sqrt{15}}{2}\sqrt{3}\sqrt{5}}{-18\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{-45\sqrt{15}}{2}\sqrt{3}\sqrt{5}}{-18\times 5}$$

Express \frac{-45\sqrt{15}}{2}\sqrt{3} as a single fraction.

$$\frac{\frac{-45\sqrt{15}\sqrt{3}}{2}\sqrt{5}}{-18\times 5}$$

Express \frac{-45\sqrt{15}\sqrt{3}}{2}\sqrt{5} as a single fraction.

$$\frac{\frac{-45\sqrt{15}\sqrt{3}\sqrt{5}}{2}}{-18\times 5}$$

Multiply -18 and 5 to get -90.

$$\frac{\frac{-45\sqrt{15}\sqrt{3}\sqrt{5}}{2}}{-90}$$

Express \frac{\frac{-45\sqrt{15}\sqrt{3}\sqrt{5}}{2}}{-90} as a single fraction.

$$\frac{-45\sqrt{15}\sqrt{3}\sqrt{5}}{2\left(-90\right)}$$

Cancel out 45 in both numerator and denominator.

$$\frac{-\sqrt{3}\sqrt{5}\sqrt{15}}{-2\times 2}$$

Cancel out -1 in both numerator and denominator.

$$\frac{\sqrt{3}\sqrt{5}\sqrt{15}}{2\times 2}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{\sqrt{3}\sqrt{5}\sqrt{3}\sqrt{5}}{2\times 2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{5}\sqrt{5}}{2\times 2}$$

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

$$\frac{3\times 5}{2\times 2}$$

Multiply 3 and 5 to get 15.

$$\frac{15}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{15}{4}$$

Factor

$$\frac{3 \cdot 5}{2 ^ {2}} = 3\frac{3}{4} = 3.75$$