Solve -5sqrt{8/27}*sqrt{11/4}*(-3sqrt{54})

$$- 5 \sqrt { \frac { 8 } { 27 } } \times \sqrt { 1 \frac { 1 } { 4 } } \times ( - 3 \sqrt { 54 } )$$

Evaluate

$$30\sqrt{5}\approx 67.082039325$$

Solution Steps

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

$$-5\times \frac{\sqrt{8}}{\sqrt{27}}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

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}.

$$-5\times \frac{2\sqrt{2}}{\sqrt{27}}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

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}.

$$-5\times \frac{2\sqrt{2}}{3\sqrt{3}}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

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

$$-5\times \frac{2\sqrt{2}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

The square of \sqrt{3} is 3.

$$-5\times \frac{2\sqrt{2}\sqrt{3}}{3\times 3}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

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

$$-5\times \frac{2\sqrt{6}}{3\times 3}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

Multiply 3 and 3 to get 9.

$$-5\times \frac{2\sqrt{6}}{9}\sqrt{\frac{4+1}{4}}\left(-3\right)\sqrt{54}$$

Add 4 and 1 to get 5.

$$-5\times \frac{2\sqrt{6}}{9}\sqrt{\frac{5}{4}}\left(-3\right)\sqrt{54}$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$-5\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{5}}{\sqrt{4}}\left(-3\right)\sqrt{54}$$

Calculate the square root of 4 and get 2.

$$-5\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{5}}{2}\left(-3\right)\sqrt{54}$$

Multiply -5 and -3 to get 15.

$$15\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{5}}{2}\sqrt{54}$$

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}.

$$15\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{5}}{2}\times 3\sqrt{6}$$

Multiply 15 and 3 to get 45.

$$45\times \frac{2\sqrt{6}}{9}\times \frac{\sqrt{5}}{2}\sqrt{6}$$

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

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

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

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

Cancel out 2 and 2.

$$5\sqrt{6}\sqrt{5}\sqrt{6}$$

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

$$5\times 6\sqrt{5}$$

Multiply 5 and 6 to get 30.

$$30\sqrt{5}$$