Solve {left(1/2right)}^2left({left(1/2right)}^2-1/2+1right)left({left(1/2right)}^3-{left(frac{1}{2}right)}^2+frac{1}{2}-1right)

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

Evaluate

$$-\frac{15}{128}=-0.1171875$$

Solution Steps

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

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

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

Since \frac{1}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 2 from 1 to get -1.

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

Convert 1 to fraction \frac{4}{4}.

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

Since -\frac{1}{4} and \frac{4}{4} have the same denominator, add them by adding their numerators.

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

Add -1 and 4 to get 3.

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

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

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

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

$$\frac{3}{16}\left(\left(\frac{1}{2}\right)^{3}-\left(\frac{1}{2}\right)^{2}+\frac{1}{2}-1\right)$$

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

$$\frac{3}{16}\left(\frac{1}{8}-\left(\frac{1}{2}\right)^{2}+\frac{1}{2}-1\right)$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{3}{16}\left(\frac{1}{8}-\frac{1}{4}+\frac{1}{2}-1\right)$$

Least common multiple of 8 and 4 is 8. Convert \frac{1}{8} and \frac{1}{4} to fractions with denominator 8.

$$\frac{3}{16}\left(\frac{1}{8}-\frac{2}{8}+\frac{1}{2}-1\right)$$

Since \frac{1}{8} and \frac{2}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{16}\left(\frac{1-2}{8}+\frac{1}{2}-1\right)$$

Subtract 2 from 1 to get -1.

$$\frac{3}{16}\left(-\frac{1}{8}+\frac{1}{2}-1\right)$$

Least common multiple of 8 and 2 is 8. Convert -\frac{1}{8} and \frac{1}{2} to fractions with denominator 8.

$$\frac{3}{16}\left(-\frac{1}{8}+\frac{4}{8}-1\right)$$

Since -\frac{1}{8} and \frac{4}{8} have the same denominator, add them by adding their numerators.

$$\frac{3}{16}\left(\frac{-1+4}{8}-1\right)$$

Add -1 and 4 to get 3.

$$\frac{3}{16}\left(\frac{3}{8}-1\right)$$

Convert 1 to fraction \frac{8}{8}.

$$\frac{3}{16}\left(\frac{3}{8}-\frac{8}{8}\right)$$

Since \frac{3}{8} and \frac{8}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{16}\times \frac{3-8}{8}$$

Subtract 8 from 3 to get -5.

$$\frac{3}{16}\left(-\frac{5}{8}\right)$$

Multiply \frac{3}{16} times -\frac{5}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(-5\right)}{16\times 8}$$

Do the multiplications in the fraction \frac{3\left(-5\right)}{16\times 8}.

$$\frac{-15}{128}$$

Fraction \frac{-15}{128} can be rewritten as -\frac{15}{128} by extracting the negative sign.

$$-\frac{15}{128}$$

Factor

$$-\frac{15}{128} = -0.1171875$$