Section 1.1
Order of Operations
There is an agreed-upon order in which arithmetic operations are to be performed.
First, compute whatever is enclosed within parentheses (and work from inner parentheses out). Then perform any exponentiation (raising to powers). Third, do all multiplications and divisions, working left to right. Fourth, do all additions and subtractions, again working left to right.
Examples:
a. \begin{align}
2+(7-3)^2 \;/\; 2&=2+4^2 \;/\; 2\\
&=2+16\;/\;2\\
&= 2+8\\
&=10
\end{align}
Note that $(7-3)^2=4^2=16$, not $7^2-3^2=49-9=40.$
b. \begin{align}
45- 12\,/\, 2\cdot 6&=45-6\cdot 6\\
&=45-36\\
&=9
\end{align}
Note that $12 \;/\; 2 \cdot 6=6\cdot 6=36$, not $12\;/\;12 =1.$
c. \begin{align}
1+2\cdot 3&=1+6\\
&=7
\end{align}
Multiplication precedes addition
d. \begin{align}
12\;/\;2\;/\;3&=6\;/3\\
&=2
\end{align}
Division is done left to right.
Exercises:
Evaluate each of the following.
1. $8-2\cdot 6$
2. $(8-2)\cdot 6$
3. $6+3/3$
4. $(6+3)/3$
5. $5-2+1$
6. $(5-2)+1$
7. $5-{{2}^{2}}+1$
8. ${{(5-2)}^{2}}+1$
9. $1+2/3-1$
10. $(1+2)/3-1$
11. $[1+2/(3-1)]$
12. $[(1+2)/(3-1)]$
13. $24/3\cdot 4/2$
14. ${{(1+2)}^{2}}-{{(3-2)}^{2}}$
15. ${{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}}$
16. ${{(1+2+3+4)}^{2}}$
17. ${{5}^{2}}-{{\left( {{3}^{2}}-{{2}^{3}} \right)}^{5}}\cdot 4\cdot 3\cdot 2$
18. ${{5}^{2}}\div {{\left( {{3}^{2}}-{{2}^{3}} \right)}^{5}}\div 5\cdot 4\cdot 3\cdot 2$
19. $1+2\cdot 3\div 4-5+6\cdot 7\div 8-9$
20. $(1+2)\cdot (3\div 4)-(5+6)\cdot 7\div (8-9)$