Solve 2^3+{left(1/3right)}^3+left|3-4right|+{left(1/sqrt{1001+1002}right)}^0

$${ 2 }^{ 3 } + { \left( \frac{ 1 }{ 3 } \right) }^{ 3 } + \left| 3-4 \right| + { \left( \frac{ 1 }{ \sqrt{ 1001 } +1002 } \right) }^{ 0 }$$

Evaluate

$$\frac{271}{27}\approx 10.037037037$$

Solution Steps

Calculate 2 to the power of 3 and get 8.

$$8+\left(\frac{1}{3}\right)^{3}+|3-4|+\left(\frac{1}{\sqrt{1001}+1002}\right)^{0}$$

Calculate \frac{1}{3} to the power of 3 and get \frac{1}{27}.

$$8+\frac{1}{27}+|3-4|+\left(\frac{1}{\sqrt{1001}+1002}\right)^{0}$$

Add 8 and \frac{1}{27} to get \frac{217}{27}.

$$\frac{217}{27}+|3-4|+\left(\frac{1}{\sqrt{1001}+1002}\right)^{0}$$

Subtract 4 from 3 to get -1.

$$\frac{217}{27}+|-1|+\left(\frac{1}{\sqrt{1001}+1002}\right)^{0}$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -1 is 1.

$$\frac{217}{27}+1+\left(\frac{1}{\sqrt{1001}+1002}\right)^{0}$$

Add \frac{217}{27} and 1 to get \frac{244}{27}.

$$\frac{244}{27}+\left(\frac{1}{\sqrt{1001}+1002}\right)^{0}$$

Rationalize the denominator of \frac{1}{\sqrt{1001}+1002} by multiplying numerator and denominator by \sqrt{1001}-1002.

$$\frac{244}{27}+\left(\frac{\sqrt{1001}-1002}{\left(\sqrt{1001}+1002\right)\left(\sqrt{1001}-1002\right)}\right)^{0}$$

Consider \left(\sqrt{1001}+1002\right)\left(\sqrt{1001}-1002\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{244}{27}+\left(\frac{\sqrt{1001}-1002}{\left(\sqrt{1001}\right)^{2}-1002^{2}}\right)^{0}$$

Square \sqrt{1001}. Square 1002.

$$\frac{244}{27}+\left(\frac{\sqrt{1001}-1002}{1001-1004004}\right)^{0}$$

Subtract 1004004 from 1001 to get -1003003.

$$\frac{244}{27}+\left(\frac{\sqrt{1001}-1002}{-1003003}\right)^{0}$$

Multiply both numerator and denominator by -1.

$$\frac{244}{27}+\left(\frac{-\sqrt{1001}+1002}{1003003}\right)^{0}$$

To raise \frac{-\sqrt{1001}+1002}{1003003} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{244}{27}+\frac{\left(-\sqrt{1001}+1002\right)^{0}}{1003003^{0}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 27 and 1003003^{0} is 27. Multiply \frac{\left(-\sqrt{1001}+1002\right)^{0}}{1003003^{0}} times \frac{27}{27}.

$$\frac{244}{27}+\frac{27\left(-\sqrt{1001}+1002\right)^{0}}{27}$$

Since \frac{244}{27} and \frac{27\left(-\sqrt{1001}+1002\right)^{0}}{27} have the same denominator, add them by adding their numerators.

$$\frac{244+27\left(-\sqrt{1001}+1002\right)^{0}}{27}$$

Calculate -\sqrt{1001}+1002 to the power of 0 and get 1.

$$\frac{244}{27}+\frac{1}{1003003^{0}}$$

Calculate 1003003 to the power of 0 and get 1.

$$\frac{244}{27}+\frac{1}{1}$$

Anything divided by one gives itself.

$$\frac{244}{27}+1$$

Add \frac{244}{27} and 1 to get \frac{271}{27}.

$$\frac{271}{27}$$

Factor

$$\frac{271}{3 ^ {3}} = 10\frac{1}{27} = 10.037037037037036$$