1.4 - Order of Operations and Invisible Brackets

Order of operations
Invisible brackets

Order of operations

An expression is a set of numbers that are combined using operations such as addition, subtraction, multiplication, division, exponentiation, etc. An example of an expression is:

4 + 5 · 2³.

In this expression the numbers are 4, 5, 2 and 3 and they are combined by the operations of addition, multiplication and exponentiation.

Expressions often use brackets ( ) as symbols of grouping. The brackets contain their own sub-expressions (thus creating expressions within expressions). An example is:

3 (4 + 5) + 2.

Here the sub-expression is 4 + 5. A more complicated example is:

10 (3 (4 + 5) + 2)

It is the previous expression multiplied by 10. This expression contains a sub-expression which itself contains a sub-expression! The question then arises: in what order do we carry out these operations? The answer is given in this table:

Order of operations:

Operation in brackets (parentheses) are done first. If there are nested brackets (brackets within brackets) then the innermost brackets are done first.
Then exponentiation.
Then multiplication and division, from left to right.
Then addition and subtraction, from left to right.

Thus the above examples would simplify like this:

Example: 4 + 5 · 2³ → 4 + 5 · 8 → 4 + 40 → 44

Example: 3 (4 + 5) + 2 → 3 · 9 + 2 → 27 + 2 → 29

Example: 10 (3 (4 + 5) + 2) → 10 (3 · 9 + 2) → 10 (27 + 2) → 10 · 29 → 290

There are a couple of acronyms that people use to help them remember the order of operations table. One is BEDMAS, which stands for the order: Brackets, Exponentiation, Division and Multiplication, Addition and Subtraction.

Another one is PEMDAS, which stands for the order: Parentheses, Exponentiation, Multiplication and Division, Addition and Subtraction. Use whichever acronym you like better.

Algebra Coach Exercises

Invisible brackets

There are three locations where brackets are usually not shown but you have to imagine that they are there. One location is the exponent in an exponential. The other two are the numerator above a horizontal division line and the denominator below a horizontal division line. Here are two examples showing expressions, first with invisible brackets, and then with visible brackets (shown in red). We also show what the expressions evaluate to.

Example 1:

In this exponential you know that the “+2” is in the exponent together with the “3” because it is written smaller and higher than the base and directly beside the “3”. This means that the exponent is meant to be “3+2” and must be kept together as if it had brackets around it.

Example 2:

In this fraction you know that the “+5” is in the numerator together with the “7” because it is written directly beside the “7” and the division line goes right across under both. The same goes for the “+4” in the denominator. This means that the numerator is “7+5” and the denominator is “2+4” and both must be kept together as if they had brackets around them.

Note:

Compare the next two examples with the two examples above. The only difference is that some of the numbers have been moved slightly or made bigger:
Conclusion: Neatness and precision count!
In the above examples we could tell that a number was in the exponent or in the denominator by how big it was or where it was written. However, when typing one of these expressions into the Algebra Coach (or any other algebra program) you can't make the numbers smaller or higher. Everything must be typed in on a single line, so you must use brackets. The following exercises show how brackets change the meaning of otherwise similar looking expressions. Try them!

Algebra Coach Exercises

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