Solve -5sqrt{8/27}*sqrt{11/3}*3sqrt{74}

$$- 5 \sqrt { \frac { 8 } { 27 } } \cdot \sqrt { 1 \frac { 1 } { 3 } } \cdot 3 \sqrt { 74 }$$

Evaluate

$$-\frac{40\sqrt{37}}{3}\approx -81.103500404$$

Solution Steps

Multiply 3 and -5 to get -15.

$$-15\sqrt{\frac{8}{27}}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

Rewrite the square root of the division \sqrt{\frac{8}{27}} as the division of square roots \frac{\sqrt{8}}{\sqrt{27}}.

$$-15\times \frac{\sqrt{8}}{\sqrt{27}}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

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}.

$$-15\times \frac{2\sqrt{2}}{\sqrt{27}}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$-15\times \frac{2\sqrt{2}}{3\sqrt{3}}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

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

$$-15\times \frac{2\sqrt{2}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

The square of \sqrt{3} is 3.

$$-15\times \frac{2\sqrt{2}\sqrt{3}}{3\times 3}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

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

$$-15\times \frac{2\sqrt{6}}{3\times 3}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

Multiply 3 and 3 to get 9.

$$-15\times \frac{2\sqrt{6}}{9}\sqrt{\frac{3+1}{3}}\sqrt{74}$$

Add 3 and 1 to get 4.

$$-15\times \frac{2\sqrt{6}}{9}\sqrt{\frac{4}{3}}\sqrt{74}$$

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

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{4}}{\sqrt{3}}\sqrt{74}$$

Calculate the square root of 4 and get 2.

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{2}{\sqrt{3}}\sqrt{74}$$

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

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{74}$$

The square of \sqrt{3} is 3.

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{2\sqrt{3}}{3}\sqrt{74}$$

Express -15\times \frac{2\sqrt{6}}{9} as a single fraction.

$$\frac{-15\times 2\sqrt{6}}{9}\times \frac{2\sqrt{3}}{3}\sqrt{74}$$

Multiply \frac{-15\times 2\sqrt{6}}{9} times \frac{2\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-15\times 2\sqrt{6}\times 2\sqrt{3}}{9\times 3}\sqrt{74}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-2\times 2\times 5\sqrt{3}\sqrt{6}}{3\times 3}\sqrt{74}$$

Express \frac{-2\times 2\times 5\sqrt{3}\sqrt{6}}{3\times 3}\sqrt{74} as a single fraction.

$$\frac{-2\times 2\times 5\sqrt{3}\sqrt{6}\sqrt{74}}{3\times 3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{-2\times 2\times 5\sqrt{3}\sqrt{3}\sqrt{2}\sqrt{74}}{3\times 3}$$

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

$$\frac{-2\times 2\times 5\times 3\sqrt{2}\sqrt{74}}{3\times 3}$$

Factor 74=2\times 37. Rewrite the square root of the product \sqrt{2\times 37} as the product of square roots \sqrt{2}\sqrt{37}.

$$\frac{-2\times 2\times 5\times 3\sqrt{2}\sqrt{2}\sqrt{37}}{3\times 3}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{-2\times 2\times 5\times 3\times 2\sqrt{37}}{3\times 3}$$

Multiply -2 and 2 to get -4.

$$\frac{-4\times 5\times 3\times 2\sqrt{37}}{3\times 3}$$

Multiply -4 and 5 to get -20.

$$\frac{-20\times 3\times 2\sqrt{37}}{3\times 3}$$

Multiply -20 and 3 to get -60.

$$\frac{-60\times 2\sqrt{37}}{3\times 3}$$

Multiply -60 and 2 to get -120.

$$\frac{-120\sqrt{37}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{-120\sqrt{37}}{9}$$

Divide -120\sqrt{37} by 9 to get -\frac{40}{3}\sqrt{37}.

$$-\frac{40}{3}\sqrt{37}$$