The order of operations is a convention that establishes the order that must be respected when performing calculations in a chain of operations.
When several operations are present in a calculation, it is called a chain of operations. The chain corresponds to a series of mathematical operations which must be carried out in a precise order according to the order of the operations.
This is the order of operations to be respected:
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Brackets
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Exponents
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Divisions and Multiplication (from left to right)
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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.
The following are two examples to help understand the steps to follow when applying the order of operations:
Example without an exponent
Begin by focusing on the brackets. Start with the most important operation in each bracket.
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In the brackets to the left, start with the multiplication.
|(8+\color{red}{2\times 2})\div(12\div4+3)| -
In the brackets to the right, divide.
|(8+4)\div(\color{red}{12\div4}+3)| -
In each bracket, end with addition.
|(\color{red}{8+4})\div(\color{red}{3+3})| -
The only operation remaining is the division.
|\color{red}{12\div6}|
|2|
Example with an exponent
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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| -
The Exponents
|8\times\color{red}{2^{3}}-4\times4\div8|
|8\times(2\times2\times2)-4\times4\div8|
|8\times8-4\times4\div8| -
Divisions and Multiplications (from left to right)
|\color{red}{8\times8}-\color{red}{4\times4}\div8|
|64-\color{red}{16\div8}|
|64-2| -
Additions and Subtractions (from left to right)
|\color{red}{64-2}|
|62|
Sometimes there are multiple levels of brackets. In this case, complete the operations inside the innermost brackets and work your way to outermost brackets.
|9^2 \div (21-18) + 7 \times \big(16 - (9 + 5)\big)^2|
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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| -
The Exponents
|\color{red}{9^2} \div 3 + 7 \times \color{red}{2^2}|
|81 \div 3 + 7 \times 4| -
Divisions and Multiplications (from left to right)
|\color{red}{81 \div 3} + 7 \times 4|
|27 + \color{red}{7 \times 4}|
|27 + 28| -
Additions and Subtractions (from left to right)
|\color{red}{27 + 28}|
|55|
The order of operations on fractions is the same as on integers. However, it is important to know the specific procedure to follow for each operation (i.e., multiplication, division, addition, and subtraction).
The following links show the procedure to follow for each operation.
||\left(\dfrac{1}{2}+\dfrac{1}{3}\div\dfrac{1}{4}\right)+ \left(\dfrac{3}{4}\times\dfrac{1}{2}\right)||
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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)| -
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}| -
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}| -
We finish by adding.
|\color{red}{\dfrac{11}{6}+\dfrac{3}{8}}|
|\color{red}{\dfrac{44}{24}+\dfrac{9}{24}}|
|\dfrac{53}{24}|