Solve -5*sqrt{8/27}*sqrt{11/3}*3sqrt{54}

$$- 5 \times \sqrt { \frac { 8 } { 27 } } \times \sqrt { 1 \frac { 1 } { 3 } } \times 3 \sqrt { 54 }$$

Evaluate

$$-40\sqrt{3}\approx -69.282032303$$

Solution Steps

Multiply 3 and -5 to get -15.

$$-15\sqrt{\frac{8}{27}}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

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

$$-15\times \frac{\sqrt{8}}{\sqrt{27}}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

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}.

$$-15\times \frac{2\sqrt{2}}{\sqrt{27}}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

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}.

$$-15\times \frac{2\sqrt{2}}{3\sqrt{3}}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

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

$$-15\times \frac{2\sqrt{2}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

The square of \sqrt{3} is 3.

$$-15\times \frac{2\sqrt{2}\sqrt{3}}{3\times 3}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

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

$$-15\times \frac{2\sqrt{6}}{3\times 3}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

Multiply 3 and 3 to get 9.

$$-15\times \frac{2\sqrt{6}}{9}\sqrt{\frac{3+1}{3}}\sqrt{54}$$

Add 3 and 1 to get 4.

$$-15\times \frac{2\sqrt{6}}{9}\sqrt{\frac{4}{3}}\sqrt{54}$$

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

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{4}}{\sqrt{3}}\sqrt{54}$$

Calculate the square root of 4 and get 2.

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{2}{\sqrt{3}}\sqrt{54}$$

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

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{54}$$

The square of \sqrt{3} is 3.

$$-15\times \frac{2\sqrt{6}}{9}\times \frac{2\sqrt{3}}{3}\sqrt{54}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

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

Multiply -15 and 3 to get -45.

$$-45\times \frac{2\sqrt{6}}{9}\times \frac{2\sqrt{3}}{3}\sqrt{6}$$

Cancel out 9, the greatest common factor in 45 and 9.

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

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

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

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

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

Multiply \sqrt{6} and \sqrt{6} to get 6.

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

Multiply -5 and 2 to get -10.

$$\frac{-10\times 6\times 2\sqrt{3}}{3}$$

Multiply -10 and 6 to get -60.

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

Multiply -60 and 2 to get -120.

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

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

$$-40\sqrt{3}$$