The Order of Operations

The order of operations is a convention that establishes the order that must be respected when performing calculations in a chain of operations.

This is the order of operations to be respected:

1. Brackets

2. Exponents

3. Divisions and Multiplication (from left to right)

4. Additions and Subtractions (from left to right)

To remember the order, we take the first letters of each of the steps and form a word: BEDMAS.

Example without an exponent

Begin by focusing on the brackets. Start with the most important operation in each bracket.

1. In the brackets to the left, start with the multiplication.
   
   |(8+\color{red}{2\times 2})\div(12\div4+3)|

2. In the brackets to the right, divide.
   
   |(8+4)\div(\color{red}{12\div4}+3)|

3. In each bracket, end with addition.
   
   |(\color{red}{8+4})\div(\color{red}{3+3})|

4. The only operation remaining is the division.
   
   |\color{red}{12\div6}|
   
   |2|

Example with an exponent

1. The Brackets
   
   |(10+\color{red}{2\times(-1)})\times2^{3}-4\times(2\times2)\div8|
   |(10+-2)\times2^{3}-4\times(\color{red}{2\times2})\div8|
   |(\color{red}{10+-2})\times2^{3}-4\times(4)\div8|
   |8\times2^{3}-4\times4\div8|

2. The Exponents
   
   |8\times\color{red}{2^{3}}-4\times4\div8|
   |8\times(2\times2\times2)-4\times4\div8|
   |8\times8-4\times4\div8|

3. Divisions and Multiplications (from left to right) 
   
   |\color{red}{8\times8}-\color{red}{4\times4}\div8|
   |64-\color{red}{16\div8}|
   |64-2|

4. Additions and Subtractions (from left to right)
   
   |\color{red}{64-2}|
   |62|

|9^2 \div (21-18) + 7 \times \big(16 - (9 + 5)\big)^2|

1. The Brackets
   
   |9^2 \div (21-18) + 7 \times \big(16 - (\color{red}{9 + 5})\big)^2|
   |9^2 \div (\color{red}{21-18}) + 7 \times (\color{red}{16 - 14})^2|
   |9^2 \div 3 + 7 \times 2^2|

2. The Exponents
   
   |\color{red}{9^2} \div 3 + 7 \times \color{red}{2^2}|
   |81 \div 3 + 7 \times 4|

3. Divisions and Multiplications (from left to right)
   
   |\color{red}{81 \div 3} + 7 \times 4|
   |27 + \color{red}{7 \times 4}|
   |27 + 28|

4. Additions and Subtractions (from left to right)
   
   |\color{red}{27 + 28}|
   |55|

||\left(\dfrac{1}{2}+\dfrac{1}{3}\div\dfrac{1}{4}\right)+ \left(\dfrac{3}{4}\times\dfrac{1}{2}\right)||

1. We begin with the operations in the brackets. Here, we must start with division in the bracket to the left.
   
   |\left(\dfrac{1}{2}+\color{red}{\dfrac{1}{3}\div\dfrac{1}{4}}\right)+\left(\dfrac{3}{4}\times\dfrac{1}{2}\right)|
   
   |\left(\dfrac{1}{2}+\color{red}{\dfrac{1}{3}\times\dfrac{4}{1}}\right)+\left(\dfrac{3}{4}\times\dfrac{1}{2}\right)|
   
   |\left(\dfrac{1}{2}+\dfrac{4}{3}\right)+\left(\dfrac{3}{4}\times\dfrac{1}{2}\right)|

2. We perform the multiplication in the bracket to the right.
   
   |\left(\dfrac{1}{2}+\dfrac{4}{3}\right)+\left(\color{red}{\dfrac{3}{4}\times\dfrac{1}{2}}\right)|
   
   |\left(\dfrac{1}{2}+\dfrac{4}{3}\right)+\dfrac{3}{8}|

3. We add the fractions in the bracket to the left.
   
   |\left(\color{red}{\dfrac{1}{2}+\dfrac{4}{3}}\right)+\dfrac{3}{8}|
   
   |\left(\color{red}{\dfrac{3}{6}+\dfrac{8}{6}}\right)+\dfrac{3}{8}|
   
   |\dfrac{11}{6}+\dfrac{3}{8}|

4. We finish by adding.
   
   |\color{red}{\dfrac{11}{6}+\dfrac{3}{8}}|
   
   |\color{red}{\dfrac{44}{24}+\dfrac{9}{24}}|
   
   |\dfrac{53}{24}|