Solve 2/3-left(1/2^3+1/2^2right)2/3+3+frac{{left(frac{1}{3}right)}^{4}}{{left(frac{1}{3}right)}^3}+left({left(frac{1}{3}right)}^0+frac{{left(frac{1}{3}right)}^3}{{left(frac{1}{3}right)}^4}right)frac{1}{2}

$$\frac{ 2 }{ 3 } - \left( \frac{ 1 }{ { 2 }^{ 3 } } + \frac{ 1 }{ { 2 }^{ 2 } } \right) \frac{ 2 }{ 3 } +3+ \frac{ { \left( \frac{ 1 }{ 3 } \right) }^{ 4 } }{ { \left( \frac{ 1 }{ 3 } \right) }^{ 3 } } + \left( { \left( \frac{ 1 }{ 3 } \right) }^{ 0 } + \frac{ { \left( \frac{ 1 }{ 3 } \right) }^{ 3 } }{ { \left( \frac{ 1 }{ 3 } \right) }^{ 4 } } \right) \frac{ 1 }{ 2 }$$

Evaluate

$$\frac{23}{4}=5.75$$

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.

$$\frac{2}{3}-\left(\frac{1}{2^{3}}+\frac{1}{2^{2}}\right)\times \frac{2}{3}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{\left(\frac{1}{3}\right)^{3}}{\left(\frac{1}{3}\right)^{4}}\right)\times \frac{1}{2}$$

Rewrite \left(\frac{1}{3}\right)^{4} as \frac{1}{3}\times \left(\frac{1}{3}\right)^{3}. Cancel out \left(\frac{1}{3}\right)^{3} in both numerator and denominator.

$$\frac{2}{3}-\left(\frac{1}{2^{3}}+\frac{1}{2^{2}}\right)\times \frac{2}{3}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{2}{3}-\left(\frac{1}{8}+\frac{1}{2^{2}}\right)\times \frac{2}{3}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2}{3}-\left(\frac{1}{8}+\frac{1}{4}\right)\times \frac{2}{3}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Add \frac{1}{8} and \frac{1}{4} to get \frac{3}{8}.

$$\frac{2}{3}-\frac{3}{8}\times \frac{2}{3}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Multiply \frac{3}{8} and \frac{2}{3} to get \frac{1}{4}.

$$\frac{2}{3}-\frac{1}{4}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Subtract \frac{1}{4} from \frac{2}{3} to get \frac{5}{12}.

$$\frac{5}{12}+3+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Add \frac{5}{12} and 3 to get \frac{41}{12}.

$$\frac{41}{12}+\left(\frac{1}{3}\right)^{1}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

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

$$\frac{41}{12}+\frac{1}{3}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Add \frac{41}{12} and \frac{1}{3} to get \frac{15}{4}.

$$\frac{15}{4}+\left(\left(\frac{1}{3}\right)^{0}+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

Calculate \frac{1}{3} to the power of 0 and get 1.

$$\frac{15}{4}+\left(1+\frac{1}{\frac{1}{3}}\right)\times \frac{1}{2}$$

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

$$\frac{15}{4}+\left(1+1\times 3\right)\times \frac{1}{2}$$

Multiply 1 and 3 to get 3.

$$\frac{15}{4}+\left(1+3\right)\times \frac{1}{2}$$

Add 1 and 3 to get 4.

$$\frac{15}{4}+4\times \frac{1}{2}$$

Multiply 4 and \frac{1}{2} to get 2.

$$\frac{15}{4}+2$$

Add \frac{15}{4} and 2 to get \frac{23}{4}.

$$\frac{23}{4}$$

Factor

$$\frac{23}{2 ^ {2}} = 5\frac{3}{4} = 5.75$$