Solve (6)sqrt{24/5}*sqrt{11/3}*(-2sqrt{5/2})

$$( 6 ) \sqrt { \frac { 24 } { 5 } } \times \sqrt { 1 \frac { 1 } { 3 } } \times ( - 2 \sqrt { \frac { 5 } { 2 } } )$$

Evaluate

$$-48$$

Solution Steps

To multiply \sqrt{\frac{24}{5}} and \sqrt{\frac{5}{2}}, multiply the numbers under the square root.

$$6\sqrt{12}\sqrt{\frac{1\times 3+1}{3}}\left(-2\right)$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

Multiply 6 and 2 to get 12.

$$12\sqrt{3}\sqrt{\frac{1\times 3+1}{3}}\left(-2\right)$$

Multiply 1 and 3 to get 3.

$$12\sqrt{3}\sqrt{\frac{3+1}{3}}\left(-2\right)$$

Add 3 and 1 to get 4.

$$12\sqrt{3}\sqrt{\frac{4}{3}}\left(-2\right)$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$12\sqrt{3}\times \frac{\sqrt{4}}{\sqrt{3}}\left(-2\right)$$

Calculate the square root of 4 and get 2.

$$12\sqrt{3}\times \frac{2}{\sqrt{3}}\left(-2\right)$$

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

$$12\sqrt{3}\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\left(-2\right)$$

The square of \sqrt{3} is 3.

$$12\sqrt{3}\times \frac{2\sqrt{3}}{3}\left(-2\right)$$

Multiply 12 and -2 to get -24.

$$-24\sqrt{3}\times \frac{2\sqrt{3}}{3}$$

Cancel out 3, the greatest common factor in 24 and 3.

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

Multiply -8 and 2 to get -16.

$$-16\sqrt{3}\sqrt{3}$$

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

$$-16\times 3$$

Multiply -16 and 3 to get -48.

$$-48$$

Factor

$$-48$$