Solve 4{left(1/2right)}^3+{left(1/2right)}^2-2(1/2)(2/3)+3{left(frac{2}{3}right)}^2-4

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

Evaluate

$$-\frac{31}{12}\approx -2.583333333$$

Solution Steps

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

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

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

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

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

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

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

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

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

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

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

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

Add 2 and 1 to get 3.

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

Cancel out 2 and 2.

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

Least common multiple of 4 and 3 is 12. Convert \frac{3}{4} and \frac{2}{3} to fractions with denominator 12.

$$\frac{9}{12}-\frac{8}{12}+3\times \left(\frac{2}{3}\right)^{2}-4$$

Since \frac{9}{12} and \frac{8}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9-8}{12}+3\times \left(\frac{2}{3}\right)^{2}-4$$

Subtract 8 from 9 to get 1.

$$\frac{1}{12}+3\times \left(\frac{2}{3}\right)^{2}-4$$

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

$$\frac{1}{12}+3\times \frac{4}{9}-4$$

Express 3\times \frac{4}{9} as a single fraction.

$$\frac{1}{12}+\frac{3\times 4}{9}-4$$

Multiply 3 and 4 to get 12.

$$\frac{1}{12}+\frac{12}{9}-4$$

Reduce the fraction \frac{12}{9} to lowest terms by extracting and canceling out 3.

$$\frac{1}{12}+\frac{4}{3}-4$$

Least common multiple of 12 and 3 is 12. Convert \frac{1}{12} and \frac{4}{3} to fractions with denominator 12.

$$\frac{1}{12}+\frac{16}{12}-4$$

Since \frac{1}{12} and \frac{16}{12} have the same denominator, add them by adding their numerators.

$$\frac{1+16}{12}-4$$

Add 1 and 16 to get 17.

$$\frac{17}{12}-4$$

Convert 4 to fraction \frac{48}{12}.

$$\frac{17}{12}-\frac{48}{12}$$

Since \frac{17}{12} and \frac{48}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{17-48}{12}$$

Subtract 48 from 17 to get -31.

$$-\frac{31}{12}$$

Factor

$$-\frac{31}{12} = -2\frac{7}{12} = -2.5833333333333335$$