Order of Operations

Suppose we wish to evaluate
2+3\times 4+3^{ 2 }+25\div 5+({ 2 }^{ 2 }-1)\times 2
To evaluate the above, we need to perform the operations in the correct order.
We do evaluate Brackets first, then we evaluate Orders which includes powers, indices, square roots, etc. Then we evaluate Division, followed by Multiplication, Addition and finally Subtraction.


This is easily remembered from the ‘word’ BODMAS, where
B stands for Brackets
O for Orders
D for Division
M for Multiplication
A for Addition and
S for Subtraction


So, for our problems we should first perform ({ 2 }^{ 2 }-1), that gives us 4-1=3
So that, we now have
2+3\times 4-{ 3 }^{ 2 }+25\div 5+3\times 2
Next, we take down the orders, we have { 3 }^{ 2 } which is 9.
So, now we have
2+3\times 4-9+5+3\times 2
After Division, comes Multiplication. So, evaluate 3\times 4=12 and 3\times 2=6
2+12-9+5+6
Following the rule of BODMAS, we now perform the addition.
2+12+5+6=25
Finally, we have 25-9=16
Therefore 2+3\times 4-{ 3 }^{ 2 }+25\div 5+({ 2 }^{ 2 }-1)\times 2=16


We notice that Addition and Subtraction are given the same priority in the order of operation. So we could have done 12-9=3in the above first and then add 2+3+5+6=16

Examples


Find the value of:
  1. 36-(5+({ 3 }^{ 2 }-2)\times 3)\div 13
  2. { (54\div 9\times 3-10) }^{ 2 }+15
  1. 36-(5+({ 3 }^{ 2 }-2)\times 3)\div 13=36-(5+7\times 3)\div 13
    • =36-26\div 13
    • =36-26\div 13
    • =36-2
    • =34
  2. { (54\div 9\times 3-10) }^{ 2 }+15={ (6\times 3-10) }^{ 2 }
    • ={ (18-10) }^{ 2 }
    • ={ 8 }^{ 2 }
    • =64

Find the value of 6+8\times 9\div 3
6+8\times 9\div 3=6+8\times 3 since 9\div 3=3 is done first.
=6+24
=30


Find the value of \frac { 1 }{ 1+\frac { 2 }{ 3 } }
First 1+\frac { 2 }{ 3 } =\frac { 3 }{ 3 } +\frac { 2 }{ 3 } =\frac { 5 }{ 3 }
3+\frac { 1 }{ (\cfrac { 3 }{ 5 } ) }
=\frac { 15 }{ 5 } +\frac { 3 }{ 5 }
=\frac { 18 }{ 5 }
=3\frac { 3 }{ 5 }


Evaluate, giving your answer as simplest as possible.

  1. 1+{ 2 }^{ 2 }-1\times 3+({ 4 }^{ 2 }\div 2\times 3)
    • Bracket first { 4 }^{ 2 }\div 2\times 3=16\div 2\times 3
    • =8\times 3
    • =24
  2. 1+{ 2 }^{ 2 }-1\times 3+24=1+4-1\times 3+24
    • =1+4-3+24
    • =26

Evaluate { { (2 }^{ 2 }-3) }^{ 100 }-({ { 2 }^{ 3 }-7) }^{ 200 }
{ { (2 }^{ 2 }-3) }^{ 100 }-({ { 2 }^{ 3 }-7) }^{ 200 }
{ 1 }^{ 100 }-{ 1 }^{ 100 }
=1-1
=0


If we have more brackets within brackets, then we perform the innermost brackets first and move outwards. For example, to evaluate
2+5-({ 2 }^{ 2 }+({ 6 }^{ 3 }\div 12))^{ 2 }\times 3
We do { 6 }^{ 3 }\div 12 first to have 216\div 12=18.
This brings us to
2+5-{ 2 }^{ 2 }+{ 18 }^{ 2 }\times 3
Next we take { 2 }^{ 2 }+18=4+18=22, so that now we have
2+5-{ 22 }^{ 2 }\times 3
Now { 2 }^{ 2 }=48450
2+5-484\times 3=7-1452
=-1445


Another example


Evaluate 3\times { 5 }^{ 2 }-12\div 4.
3\times { 5 }^{ 2 }-12\div 4=3\times 25-12\div 4 (Power first); { 5 }^{ 2 }=25
=3\times 25-3 (Division, 12\div 4=3)
=75-3 (Multiplication)
=72