Solve (2sqrt{2/3}-5sqrt{3/8}+4sqrt{3/2})sqrt{2/3}

$$(2 \sqrt{ \frac{ 2 }{ 3 } } -5 \sqrt{ \frac{ 3 }{ 8 } } +4 \sqrt{ \frac{ 3 }{ 2 } } ) \sqrt{ \frac{ 2 }{ 3 } }$$

Evaluate

$$\frac{17}{6}\approx 2.833333333$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\left(2\times \frac{\sqrt{2}}{\sqrt{3}}-5\sqrt{\frac{3}{8}}+4\sqrt{\frac{3}{2}}\right)\sqrt{\frac{2}{3}}$$

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

$$\left(2\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-5\sqrt{\frac{3}{8}}+4\sqrt{\frac{3}{2}}\right)\sqrt{\frac{2}{3}}$$

The square of \sqrt{3} is 3.

$$\left(2\times \frac{\sqrt{2}\sqrt{3}}{3}-5\sqrt{\frac{3}{8}}+4\sqrt{\frac{3}{2}}\right)\sqrt{\frac{2}{3}}$$

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

$$\left(2\times \frac{\sqrt{6}}{3}-5\sqrt{\frac{3}{8}}+4\sqrt{\frac{3}{2}}\right)\sqrt{\frac{2}{3}}$$

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

$$\left(\frac{2\sqrt{6}}{3}-5\sqrt{\frac{3}{8}}+4\sqrt{\frac{3}{2}}\right)\sqrt{\frac{2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{3}{8}} as the division of square roots \frac{\sqrt{3}}{\sqrt{8}}.

$$\left(\frac{2\sqrt{6}}{3}-5\times \frac{\sqrt{3}}{\sqrt{8}}+4\sqrt{\frac{3}{2}}\right)\sqrt{\frac{2}{3}}$$

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}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply 2 and 2 to get 4.

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

Express -5\times \frac{\sqrt{6}}{4} as a single fraction.

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

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Cancel out 2, the greatest common factor in 4 and 2.

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

Combine \frac{2\sqrt{6}}{3} and 2\sqrt{6} to get \frac{8}{3}\sqrt{6}.

$$\left(\frac{8}{3}\sqrt{6}+\frac{-5\sqrt{6}}{4}\right)\sqrt{\frac{2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\left(\frac{8}{3}\sqrt{6}+\frac{-5\sqrt{6}}{4}\right)\times \frac{\sqrt{2}}{\sqrt{3}}$$

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

$$\left(\frac{8}{3}\sqrt{6}+\frac{-5\sqrt{6}}{4}\right)\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\left(\frac{8}{3}\sqrt{6}+\frac{-5\sqrt{6}}{4}\right)\times \frac{\sqrt{2}\sqrt{3}}{3}$$

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

$$\left(\frac{8}{3}\sqrt{6}+\frac{-5\sqrt{6}}{4}\right)\times \frac{\sqrt{6}}{3}$$

Use the distributive property to multiply \frac{8}{3}\sqrt{6}+\frac{-5\sqrt{6}}{4} by \frac{\sqrt{6}}{3}.

$$\frac{8}{3}\sqrt{6}\times \frac{\sqrt{6}}{3}+\frac{-5\sqrt{6}}{4}\times \frac{\sqrt{6}}{3}$$

Multiply \frac{8}{3} times \frac{\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8\sqrt{6}}{3\times 3}\sqrt{6}+\frac{-5\sqrt{6}}{4}\times \frac{\sqrt{6}}{3}$$

Multiply \frac{-5\sqrt{6}}{4} times \frac{\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8\sqrt{6}}{3\times 3}\sqrt{6}+\frac{-5\sqrt{6}\sqrt{6}}{4\times 3}$$

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

$$\frac{8\sqrt{6}}{3\times 3}\sqrt{6}+\frac{-5\times 6}{4\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{8\sqrt{6}}{9}\sqrt{6}+\frac{-5\times 6}{4\times 3}$$

Express \frac{8\sqrt{6}}{9}\sqrt{6} as a single fraction.

$$\frac{8\sqrt{6}\sqrt{6}}{9}+\frac{-5\times 6}{4\times 3}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{8\sqrt{6}\sqrt{6}}{9}+\frac{-5}{2}$$

Fraction \frac{-5}{2} can be rewritten as -\frac{5}{2} by extracting the negative sign.

$$\frac{8\sqrt{6}\sqrt{6}}{9}-\frac{5}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 2 is 18. Multiply \frac{8\sqrt{6}\sqrt{6}}{9} times \frac{2}{2}. Multiply \frac{5}{2} times \frac{9}{9}.

$$\frac{2\times 8\sqrt{6}\sqrt{6}}{18}-\frac{5\times 9}{18}$$

Since \frac{2\times 8\sqrt{6}\sqrt{6}}{18} and \frac{5\times 9}{18} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 8\sqrt{6}\sqrt{6}-5\times 9}{18}$$

Do the multiplications in 2\times 8\sqrt{6}\sqrt{6}-5\times 9.

$$\frac{96-45}{18}$$

Do the calculations in 96-45.

$$\frac{51}{18}$$

Reduce the fraction \frac{51}{18} to lowest terms by extracting and canceling out 3.

$$\frac{17}{6}$$

Factor

$$\frac{17}{2 \cdot 3} = 2\frac{5}{6} = 2.8333333333333335$$