At numeric expressions are sets of numbers and math operations where the order of these operations is well defined so that there is a convention regarding their result. The operations involved in numeric expressions are the basics of mathematics: addition, subtraction, multiplication, division, potentiation and rooting. Below is an example of a numeric expression:
[(3·5 + 4) – (21·31)]·7
There is one order which must be followed for the solution of the whole numeric expression. Note below which operations must be done first:
Ordering of operations
1 – Potentiations or radiciations. These operations should be the first to be done. Between these two there is no priority, so they can be calculated as best.
2 – Multiplications or divisions. In cases where the enhancements and rootings have already been done or do not exist, the sequence of operations to be calculated is multiplications or divisions. There is also no priority between them, so multiplying first or dividing first is up to the person who calculates.
3 – Additions and subtractions. These are the last to be done in the priority ranking of numeric expressions. They can also be made in any order.
For example, look at the resolution of the numeric expression below, in which the order given above was applied.
4 + 2·72 – 49
First, potentiation or rooting.
4 + 2·49 – 49
Second, multiplications or divisions.
4 + 98 – 49
Thirdly, additions and subtractions. We'll do additions between numbers that have the same sign first, then additions between numbers with different signs. The properties used for this come from adding whole numbers.
102 – 49
53
special ordering
within the numeric expressions it is possible that some operations are placed with higher priority than others, even though in the order given above they have a lower priority. This new priority is given by the use of parentheses, square brackets, and braces.
Thus, the new priority for numeric expressions, when these have parentheses, square brackets and braces, it is as follows:
1 - Parenses. First, the operations that are enclosed in parentheses must be done before all others. Operations inside parentheses must be done at the priority discussed earlier.
2 – brackets. Second, the operations that are enclosed in square brackets must be performed. They should also follow the priority of basic math operations.
3 – Keys. Thirdly, the operations that remain within the keys must be calculated, also in the same order discussed above.
4 – Perform operations that remain out of keys.
Just remember that if only one number is left inside the parentheses, they can be eliminated. The same goes for braces and square brackets. Look at the example below, involving the special ordering just described and the order of operations already discussed.
{[(2 + 5·3)·2 – 7]·10 + 1} + 16
First, perform the calculations inside the parentheses and eliminate them. As the priority for its interior is multiplication, we will have:
{[(2 + 15)·2 – 7]·10 + 1} + 16
{[17·2 – 7]·10 + 1} + 16
Now perform the calculations inside the square brackets and eliminate them. We will also have a multiplication to be performed with priority.
{[34 – 7]·10 + 1} + 16
{27·10 + 1} + 16
Finally, do the calculations inside the square brackets, eliminate them and do the remaining calculations.
{270 + 1} + 16
271 + 16
287
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-expressao-numerica.htm
