Solve sqrt{frac{1/16}*2/5}{6/5*1/2}+frac{1}{100}-frac{1}{20}sqrt{frac{3}{4}-frac{1}{2}}+15(frac{1}{10})^3

$$\frac { \sqrt { \frac { 1 } { 16 } } \times \frac { 2 } { 5 } } { \frac { 6 } { 5 } \times \frac { 1 } { 2 } } + \frac { 1 } { 100 } - \frac { 1 } { 20 } \sqrt { \frac { 3 } { 4 } - \frac { 1 } { 2 } } + 15 ( \frac { 1 } { 10 } ) ^ { 3 }$$

Evaluate

$$\frac{1}{6}\approx 0.166666667$$

Solution Steps

Rewrite the square root of the division \frac{1}{16} as the division of square roots \frac{\sqrt{1}}{\sqrt{16}}. Take the square root of both numerator and denominator.

$$\frac{\frac{1}{4}\times \frac{2}{5}}{\frac{6}{5}\times \frac{1}{2}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Multiply \frac{1}{4} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{1\times 2}{4\times 5}}{\frac{6}{5}\times \frac{1}{2}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Do the multiplications in the fraction \frac{1\times 2}{4\times 5}.

$$\frac{\frac{2}{20}}{\frac{6}{5}\times \frac{1}{2}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Reduce the fraction \frac{2}{20} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{1}{10}}{\frac{6}{5}\times \frac{1}{2}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Multiply \frac{6}{5} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{1}{10}}{\frac{6\times 1}{5\times 2}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Do the multiplications in the fraction \frac{6\times 1}{5\times 2}.

$$\frac{\frac{1}{10}}{\frac{6}{10}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

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

$$\frac{\frac{1}{10}}{\frac{3}{5}}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Divide \frac{1}{10} by \frac{3}{5} by multiplying \frac{1}{10} by the reciprocal of \frac{3}{5}.

$$\frac{1}{10}\times \frac{5}{3}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Multiply \frac{1}{10} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 5}{10\times 3}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Do the multiplications in the fraction \frac{1\times 5}{10\times 3}.

$$\frac{5}{30}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Reduce the fraction \frac{5}{30} to lowest terms by extracting and canceling out 5.

$$\frac{1}{6}+\frac{1}{100}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Least common multiple of 6 and 100 is 300. Convert \frac{1}{6} and \frac{1}{100} to fractions with denominator 300.

$$\frac{50}{300}+\frac{3}{300}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Since \frac{50}{300} and \frac{3}{300} have the same denominator, add them by adding their numerators.

$$\frac{50+3}{300}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Add 50 and 3 to get 53.

$$\frac{53}{300}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{1}{2}}+15\times \left(\frac{1}{10}\right)^{3}$$

Least common multiple of 4 and 2 is 4. Convert \frac{3}{4} and \frac{1}{2} to fractions with denominator 4.

$$\frac{53}{300}-\frac{1}{20}\sqrt{\frac{3}{4}-\frac{2}{4}}+15\times \left(\frac{1}{10}\right)^{3}$$

Since \frac{3}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{53}{300}-\frac{1}{20}\sqrt{\frac{3-2}{4}}+15\times \left(\frac{1}{10}\right)^{3}$$

Subtract 2 from 3 to get 1.

$$\frac{53}{300}-\frac{1}{20}\sqrt{\frac{1}{4}}+15\times \left(\frac{1}{10}\right)^{3}$$

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.

$$\frac{53}{300}-\frac{1}{20}\times \frac{1}{2}+15\times \left(\frac{1}{10}\right)^{3}$$

Multiply \frac{1}{20} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{53}{300}-\frac{1\times 1}{20\times 2}+15\times \left(\frac{1}{10}\right)^{3}$$

Do the multiplications in the fraction \frac{1\times 1}{20\times 2}.

$$\frac{53}{300}-\frac{1}{40}+15\times \left(\frac{1}{10}\right)^{3}$$

Least common multiple of 300 and 40 is 600. Convert \frac{53}{300} and \frac{1}{40} to fractions with denominator 600.

$$\frac{106}{600}-\frac{15}{600}+15\times \left(\frac{1}{10}\right)^{3}$$

Since \frac{106}{600} and \frac{15}{600} have the same denominator, subtract them by subtracting their numerators.

$$\frac{106-15}{600}+15\times \left(\frac{1}{10}\right)^{3}$$

Subtract 15 from 106 to get 91.

$$\frac{91}{600}+15\times \left(\frac{1}{10}\right)^{3}$$

Calculate \frac{1}{10} to the power of 3 and get \frac{1}{1000}.

$$\frac{91}{600}+15\times \frac{1}{1000}$$

Multiply 15 and \frac{1}{1000} to get \frac{15}{1000}.

$$\frac{91}{600}+\frac{15}{1000}$$

Reduce the fraction \frac{15}{1000} to lowest terms by extracting and canceling out 5.

$$\frac{91}{600}+\frac{3}{200}$$

Least common multiple of 600 and 200 is 600. Convert \frac{91}{600} and \frac{3}{200} to fractions with denominator 600.

$$\frac{91}{600}+\frac{9}{600}$$

Since \frac{91}{600} and \frac{9}{600} have the same denominator, add them by adding their numerators.

$$\frac{91+9}{600}$$

Add 91 and 9 to get 100.

$$\frac{100}{600}$$

Reduce the fraction \frac{100}{600} to lowest terms by extracting and canceling out 100.

$$\frac{1}{6}$$

Factor

$$\frac{1}{2 \cdot 3} = 0.16666666666666666$$