Solve 8*(-frac{1/2)^{2}-16*(-1/2)^{3}*(2/3)^{2}+8*(-1/2)*(frac{2}{3})4}{(-frac{1}{2})^3+3*(-frac{1}{2})*(frac{2}{3})^2+(frac{2}{3})^3+3*(frac{2}{3})*(-frac{1}{2})^3}

$$\frac { 8 \times ( - \frac { 1 } { 2 } ) ^ { 2 } - 16 \times ( - \frac { 1 } { 2 } ) ^ { 3 } \times ( \frac { 2 } { 3 } ) ^ { 2 } + 8 \times ( - \frac { 1 } { 2 } ) \times ( \frac { 2 } { 3 } ) 4 } { ( - \frac { 1 } { 2 } ) ^ { 3 } + 3 \times ( - \frac { 1 } { 2 } ) \times ( \frac { 2 } { 3 } ) ^ { 2 } + ( \frac { 2 } { 3 } ) ^ { 3 } + 3 \times ( \frac { 2 } { 3 } ) \times ( - \frac { 1 } { 2 } ) ^ { 3 } }$$

Evaluate

$$\frac{240}{23}\approx 10.434782609$$

Solution Steps

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

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

Multiply 8 and \frac{1}{4} to get \frac{8}{4}.

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

Divide 8 by 4 to get 2.

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

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

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

Express 16\left(-\frac{1}{8}\right) as a single fraction.

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

Multiply 16 and -1 to get -16.

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

Divide -16 by 8 to get -2.

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

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

$$\frac{2-\left(-2\times \frac{4}{9}\right)+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Express -2\times \frac{4}{9} as a single fraction.

$$\frac{2-\frac{-2\times 4}{9}+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Multiply -2 and 4 to get -8.

$$\frac{2-\frac{-8}{9}+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Fraction \frac{-8}{9} can be rewritten as -\frac{8}{9} by extracting the negative sign.

$$\frac{2-\left(-\frac{8}{9}\right)+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

The opposite of -\frac{8}{9} is \frac{8}{9}.

$$\frac{2+\frac{8}{9}+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Convert 2 to fraction \frac{18}{9}.

$$\frac{\frac{18}{9}+\frac{8}{9}+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Since \frac{18}{9} and \frac{8}{9} have the same denominator, add them by adding their numerators.

$$\frac{\frac{18+8}{9}+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Add 18 and 8 to get 26.

$$\frac{\frac{26}{9}+8\left(-\frac{1}{2}\right)\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Express 8\left(-\frac{1}{2}\right) as a single fraction.

$$\frac{\frac{26}{9}+\frac{8\left(-1\right)}{2}\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Multiply 8 and -1 to get -8.

$$\frac{\frac{26}{9}+\frac{-8}{2}\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Divide -8 by 2 to get -4.

$$\frac{\frac{26}{9}-4\times \frac{2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Express -4\times \frac{2}{3} as a single fraction.

$$\frac{\frac{26}{9}+\frac{-4\times 2}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Multiply -4 and 2 to get -8.

$$\frac{\frac{26}{9}+\frac{-8}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Fraction \frac{-8}{3} can be rewritten as -\frac{8}{3} by extracting the negative sign.

$$\frac{\frac{26}{9}-\frac{8}{3}\times 4}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Express -\frac{8}{3}\times 4 as a single fraction.

$$\frac{\frac{26}{9}+\frac{-8\times 4}{3}}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Multiply -8 and 4 to get -32.

$$\frac{\frac{26}{9}+\frac{-32}{3}}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Fraction \frac{-32}{3} can be rewritten as -\frac{32}{3} by extracting the negative sign.

$$\frac{\frac{26}{9}-\frac{32}{3}}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Least common multiple of 9 and 3 is 9. Convert \frac{26}{9} and \frac{32}{3} to fractions with denominator 9.

$$\frac{\frac{26}{9}-\frac{96}{9}}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Since \frac{26}{9} and \frac{96}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{26-96}{9}}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Subtract 96 from 26 to get -70.

$$\frac{-\frac{70}{9}}{\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

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

$$\frac{-\frac{70}{9}}{-\frac{1}{8}+3\left(-\frac{1}{2}\right)\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Express 3\left(-\frac{1}{2}\right) as a single fraction.

$$\frac{-\frac{70}{9}}{-\frac{1}{8}+\frac{3\left(-1\right)}{2}\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Multiply 3 and -1 to get -3.

$$\frac{-\frac{70}{9}}{-\frac{1}{8}+\frac{-3}{2}\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

$$\frac{-\frac{70}{9}}{-\frac{1}{8}-\frac{3}{2}\times \left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

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

$$\frac{-\frac{70}{9}}{-\frac{1}{8}-\frac{3}{2}\times \frac{4}{9}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Multiply -\frac{3}{2} times \frac{4}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\frac{70}{9}}{-\frac{1}{8}+\frac{-3\times 4}{2\times 9}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Do the multiplications in the fraction \frac{-3\times 4}{2\times 9}.

$$\frac{-\frac{70}{9}}{-\frac{1}{8}+\frac{-12}{18}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Reduce the fraction \frac{-12}{18} to lowest terms by extracting and canceling out 6.

$$\frac{-\frac{70}{9}}{-\frac{1}{8}-\frac{2}{3}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Least common multiple of 8 and 3 is 24. Convert -\frac{1}{8} and \frac{2}{3} to fractions with denominator 24.

$$\frac{-\frac{70}{9}}{-\frac{3}{24}-\frac{16}{24}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Since -\frac{3}{24} and \frac{16}{24} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\frac{70}{9}}{\frac{-3-16}{24}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Subtract 16 from -3 to get -19.

$$\frac{-\frac{70}{9}}{-\frac{19}{24}+\left(\frac{2}{3}\right)^{3}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Calculate \frac{2}{3} to the power of 3 and get \frac{8}{27}.

$$\frac{-\frac{70}{9}}{-\frac{19}{24}+\frac{8}{27}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Least common multiple of 24 and 27 is 216. Convert -\frac{19}{24} and \frac{8}{27} to fractions with denominator 216.

$$\frac{-\frac{70}{9}}{-\frac{171}{216}+\frac{64}{216}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Since -\frac{171}{216} and \frac{64}{216} have the same denominator, add them by adding their numerators.

$$\frac{-\frac{70}{9}}{\frac{-171+64}{216}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Add -171 and 64 to get -107.

$$\frac{-\frac{70}{9}}{-\frac{107}{216}+3\times \frac{2}{3}\left(-\frac{1}{2}\right)^{3}}$$

Cancel out 3 and 3.

$$\frac{-\frac{70}{9}}{-\frac{107}{216}+2\left(-\frac{1}{2}\right)^{3}}$$

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

$$\frac{-\frac{70}{9}}{-\frac{107}{216}+2\left(-\frac{1}{8}\right)}$$

Express 2\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{-\frac{70}{9}}{-\frac{107}{216}+\frac{2\left(-1\right)}{8}}$$

Multiply 2 and -1 to get -2.

$$\frac{-\frac{70}{9}}{-\frac{107}{216}+\frac{-2}{8}}$$

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

$$\frac{-\frac{70}{9}}{-\frac{107}{216}-\frac{1}{4}}$$

Least common multiple of 216 and 4 is 216. Convert -\frac{107}{216} and \frac{1}{4} to fractions with denominator 216.

$$\frac{-\frac{70}{9}}{-\frac{107}{216}-\frac{54}{216}}$$

Since -\frac{107}{216} and \frac{54}{216} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\frac{70}{9}}{\frac{-107-54}{216}}$$

Subtract 54 from -107 to get -161.

$$\frac{-\frac{70}{9}}{-\frac{161}{216}}$$

Divide -\frac{70}{9} by -\frac{161}{216} by multiplying -\frac{70}{9} by the reciprocal of -\frac{161}{216}.

$$-\frac{70}{9}\left(-\frac{216}{161}\right)$$

Multiply -\frac{70}{9} times -\frac{216}{161} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-70\left(-216\right)}{9\times 161}$$

Do the multiplications in the fraction \frac{-70\left(-216\right)}{9\times 161}.

$$\frac{15120}{1449}$$

Reduce the fraction \frac{15120}{1449} to lowest terms by extracting and canceling out 63.

$$\frac{240}{23}$$

Factor

$$\frac{2 ^ {4} \cdot 3 \cdot 5}{23} = 10\frac{10}{23} = 10.434782608695652$$