Solve (1/2+(1/2+(1/2+2/6):frac{10}{8})):(frac{3}{2})^2-(frac{1}{2})^4:(frac{1}{2})^3

$$( \frac { 1 } { 2 } + ( \frac { 1 } { 2 } + ( \frac { 1 } { 2 } + \frac { 2 } { 6 } ) : \frac { 10 } { 8 } ) ) : ( \frac { 3 } { 2 } ) ^ { 2 } - ( \frac { 1 } { 2 } ) ^ { 4 } : ( \frac { 1 } { 2 } ) ^ { 3 }$$

Evaluate

$$\frac{13}{54}\approx 0.240740741$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 4 to get 1.

$$\frac{\frac{1}{2}+\frac{1}{2}+\frac{\frac{1}{2}+\frac{2}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

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

$$\frac{\frac{1+1}{2}+\frac{\frac{1}{2}+\frac{2}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Add 1 and 1 to get 2.

$$\frac{\frac{2}{2}+\frac{\frac{1}{2}+\frac{2}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Divide 2 by 2 to get 1.

$$\frac{1+\frac{\frac{1}{2}+\frac{2}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

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

$$\frac{1+\frac{\frac{1}{2}+\frac{1}{3}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

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

$$\frac{1+\frac{\frac{3}{6}+\frac{2}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{1+\frac{\frac{3+2}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Add 3 and 2 to get 5.

$$\frac{1+\frac{\frac{5}{6}}{\frac{10}{8}}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Reduce the fraction \frac{10}{8} to lowest terms by extracting and canceling out 2.

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

Divide \frac{5}{6} by \frac{5}{4} by multiplying \frac{5}{6} by the reciprocal of \frac{5}{4}.

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

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

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

Cancel out 5 in both numerator and denominator.

$$\frac{1+\frac{4}{6}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1+\frac{2}{3}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{\frac{3}{3}+\frac{2}{3}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Since \frac{3}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3+2}{3}}{\left(\frac{3}{2}\right)^{2}}-\left(\frac{1}{2}\right)^{1}$$

Add 3 and 2 to get 5.

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

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{\frac{5}{3}}{\frac{9}{4}}-\left(\frac{1}{2}\right)^{1}$$

Divide \frac{5}{3} by \frac{9}{4} by multiplying \frac{5}{3} by the reciprocal of \frac{9}{4}.

$$\frac{5}{3}\times \frac{4}{9}-\left(\frac{1}{2}\right)^{1}$$

Multiply \frac{5}{3} times \frac{4}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 4}{3\times 9}-\left(\frac{1}{2}\right)^{1}$$

Do the multiplications in the fraction \frac{5\times 4}{3\times 9}.

$$\frac{20}{27}-\left(\frac{1}{2}\right)^{1}$$

Calculate \frac{1}{2} to the power of 1 and get \frac{1}{2}.

$$\frac{20}{27}-\frac{1}{2}$$

Least common multiple of 27 and 2 is 54. Convert \frac{20}{27} and \frac{1}{2} to fractions with denominator 54.

$$\frac{40}{54}-\frac{27}{54}$$

Since \frac{40}{54} and \frac{27}{54} have the same denominator, subtract them by subtracting their numerators.

$$\frac{40-27}{54}$$

Subtract 27 from 40 to get 13.

$$\frac{13}{54}$$

Factor

$$\frac{13}{2 \cdot 3 ^ {3}} = 0.24074074074074073$$