Solve left(5/2sqrt{2}-3sqrt{6}right)left(1/2sqrt{2}-1/3sqrt{6}right)

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

Evaluate

$$-\frac{14\sqrt{3}}{3}+\frac{17}{2}\approx 0.417096231$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{5}{2}\sqrt{2}-3\sqrt{6} by each term of \frac{1}{2}\sqrt{2}-\frac{1}{3}\sqrt{6}.

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

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

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

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

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

Cancel out 2 and 2.

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

Multiply 5 and \frac{1}{2} to get \frac{5}{2}.

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

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

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

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

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

Cancel out 2 and 2.

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

Express 5\left(-\frac{1}{3}\right) as a single fraction.

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

Multiply 5 and -1 to get -5.

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

Fraction \frac{-5}{3} can be rewritten as -\frac{5}{3} by extracting the negative sign.

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

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

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

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

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

Multiply -3 and 2 to get -6.

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

Multiply -6 and \frac{1}{2} to get \frac{-6}{2}.

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

Divide -6 by 2 to get -3.

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

Combine -\frac{5}{3}\sqrt{3} and -3\sqrt{3} to get -\frac{14}{3}\sqrt{3}.

$$\frac{5}{2}-\frac{14}{3}\sqrt{3}-3\times 6\left(-\frac{1}{3}\right)$$

Multiply -3 and 6 to get -18.

$$\frac{5}{2}-\frac{14}{3}\sqrt{3}-18\left(-\frac{1}{3}\right)$$

Express -18\left(-\frac{1}{3}\right) as a single fraction.

$$\frac{5}{2}-\frac{14}{3}\sqrt{3}+\frac{-18\left(-1\right)}{3}$$

Multiply -18 and -1 to get 18.

$$\frac{5}{2}-\frac{14}{3}\sqrt{3}+\frac{18}{3}$$

Divide 18 by 3 to get 6.

$$\frac{5}{2}-\frac{14}{3}\sqrt{3}+6$$

Convert 6 to fraction \frac{12}{2}.

$$\frac{5}{2}-\frac{14}{3}\sqrt{3}+\frac{12}{2}$$

Since \frac{5}{2} and \frac{12}{2} have the same denominator, add them by adding their numerators.

$$\frac{5+12}{2}-\frac{14}{3}\sqrt{3}$$

Add 5 and 12 to get 17.

$$\frac{17}{2}-\frac{14}{3}\sqrt{3}$$

Factor

$$\frac{51 - 28 \sqrt{3}}{6} = 0.4170962313452395$$