Solve sqrt{1/42/3+1/8/frac{3{2}}}sqrt{left(frac{5}{18}+frac{1}{54}right)left(frac{5}{2}-1right)}+left(frac{1}{30}+frac{1}{6}right)left(frac{3}{2}+frac{1}{6}right)

$$\sqrt{ \frac{ 1 }{ 4 } \frac{ 2 }{ 3 } + \frac{ \frac{ 1 }{ 8 } }{ \frac{ 3 }{ 2 } } } \sqrt{ \left( \frac{ 5 }{ 18 } + \frac{ 1 }{ 54 } \right) \left( \frac{ 5 }{ 2 } -1 \right) } + \left( \frac{ 1 }{ 30 } + \frac{ 1 }{ 6 } \right) \left( \frac{ 3 }{ 2 } + \frac{ 1 }{ 6 } \right)$$

Evaluate

$$\frac{2}{3}\approx 0.666666667$$

Solution Steps

Multiply \frac{1}{4} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1\times 2}{4\times 3}+\frac{\frac{1}{8}}{\frac{3}{2}}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Do the multiplications in the fraction \frac{1\times 2}{4\times 3}.

$$\sqrt{\frac{2}{12}+\frac{\frac{1}{8}}{\frac{3}{2}}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1}{6}+\frac{\frac{1}{8}}{\frac{3}{2}}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Divide \frac{1}{8} by \frac{3}{2} by multiplying \frac{1}{8} by the reciprocal of \frac{3}{2}.

$$\sqrt{\frac{1}{6}+\frac{1}{8}\times \frac{2}{3}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Multiply \frac{1}{8} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1}{6}+\frac{1\times 2}{8\times 3}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Do the multiplications in the fraction \frac{1\times 2}{8\times 3}.

$$\sqrt{\frac{1}{6}+\frac{2}{24}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Reduce the fraction \frac{2}{24} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1}{6}+\frac{1}{12}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Least common multiple of 6 and 12 is 12. Convert \frac{1}{6} and \frac{1}{12} to fractions with denominator 12.

$$\sqrt{\frac{2}{12}+\frac{1}{12}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Since \frac{2}{12} and \frac{1}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{2+1}{12}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Add 2 and 1 to get 3.

$$\sqrt{\frac{3}{12}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Reduce the fraction \frac{3}{12} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{1}{4}}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Rewrite the square root of the division \frac{1}{4} as the division of square roots \frac{\sqrt{1}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{1}{2}\sqrt{\left(\frac{5}{18}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Least common multiple of 18 and 54 is 54. Convert \frac{5}{18} and \frac{1}{54} to fractions with denominator 54.

$$\frac{1}{2}\sqrt{\left(\frac{15}{54}+\frac{1}{54}\right)\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Since \frac{15}{54} and \frac{1}{54} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}\sqrt{\frac{15+1}{54}\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Add 15 and 1 to get 16.

$$\frac{1}{2}\sqrt{\frac{16}{54}\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Reduce the fraction \frac{16}{54} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}\sqrt{\frac{8}{27}\left(\frac{5}{2}-1\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{1}{2}\sqrt{\frac{8}{27}\left(\frac{5}{2}-\frac{2}{2}\right)}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Since \frac{5}{2} and \frac{2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2}\sqrt{\frac{8}{27}\times \frac{5-2}{2}}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Subtract 2 from 5 to get 3.

$$\frac{1}{2}\sqrt{\frac{8}{27}\times \frac{3}{2}}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Multiply \frac{8}{27} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{2}\sqrt{\frac{8\times 3}{27\times 2}}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Do the multiplications in the fraction \frac{8\times 3}{27\times 2}.

$$\frac{1}{2}\sqrt{\frac{24}{54}}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Reduce the fraction \frac{24}{54} to lowest terms by extracting and canceling out 6.

$$\frac{1}{2}\sqrt{\frac{4}{9}}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Rewrite the square root of the division \frac{4}{9} as the division of square roots \frac{\sqrt{4}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$\frac{1}{2}\times \frac{2}{3}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Multiply \frac{1}{2} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2}{2\times 3}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{3}+\left(\frac{1}{30}+\frac{1}{6}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Least common multiple of 30 and 6 is 30. Convert \frac{1}{30} and \frac{1}{6} to fractions with denominator 30.

$$\frac{1}{3}+\left(\frac{1}{30}+\frac{5}{30}\right)\left(\frac{3}{2}+\frac{1}{6}\right)$$

Since \frac{1}{30} and \frac{5}{30} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}+\frac{1+5}{30}\left(\frac{3}{2}+\frac{1}{6}\right)$$

Add 1 and 5 to get 6.

$$\frac{1}{3}+\frac{6}{30}\left(\frac{3}{2}+\frac{1}{6}\right)$$

Reduce the fraction \frac{6}{30} to lowest terms by extracting and canceling out 6.

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

Least common multiple of 2 and 6 is 6. Convert \frac{3}{2} and \frac{1}{6} to fractions with denominator 6.

$$\frac{1}{3}+\frac{1}{5}\left(\frac{9}{6}+\frac{1}{6}\right)$$

Since \frac{9}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}+\frac{1}{5}\times \frac{9+1}{6}$$

Add 9 and 1 to get 10.

$$\frac{1}{3}+\frac{1}{5}\times \frac{10}{6}$$

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{3}+\frac{1}{5}\times \frac{5}{3}$$

Multiply \frac{1}{5} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}+\frac{1\times 5}{5\times 3}$$

Cancel out 5 in both numerator and denominator.

$$\frac{1}{3}+\frac{1}{3}$$

Since \frac{1}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$\frac{1+1}{3}$$

Add 1 and 1 to get 2.

$$\frac{2}{3}$$

Factor

$$\frac{2}{3} = 0.6666666666666666$$