Order of Operations

If you are asked to simplify something like 5 + 7 × 2, the question that arises is which way do I do this?

Do you solve as (5 + 7) × 2 = 24 or 5 + (7 × 2) = 19?

It may seem that the first answer is correct because you worked from left to right but we don’t know for sure.  However mathematics is precise and we can’t have this kind of flexibility so, naturally, there are rules that determine which calculations take precedence.

Probably the most popular technique for remembering the order of operations is the phrase: Please Excuse MDear Aunt Sally which stands for Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.

This rule tells you that generally, parentheses and exponents outrank multiplication and division, and they outrank addition and subtraction.  In case you’re wondering, they’re are paired off because they are of equal rank.  When you have operations of the same rank, you work from left to right.  For instance, 24 ÷ 3 × 4 ≠ 2; it’s 32 because, going from left to right, you do the division first.  If you’re using a calculator, it’s been programmed with the order of operations hierarchy.

So, by applying the order of operations to our original problem, we see that 5 + (7 × 2) = 19 is correct because we have to do the multiplication before the addition.

But PEMDAS can cause some confusion.  Many times you must work problems from the inside out, because some parts of the problem are nested inside other parts.  The best way to explain this is to do some examples:

Simplify 5 + (2 + 3)²

You have to simplify inside the parentheses before applying the exponent:

5 + (2 + 3)² = 5 + (5)² = 5 + 25 = 30

Simplify 8 + [−2(−3 − 2)]²

This is a case where you must work from the inside out:

8 + [−2(−3 − 2)]² =
=
=
=
8 + [−2(−5)]²
8 + [10]²
8 + 100
108
Note that there is no rule concerning the use of parentheses, brackets or braces; they are used when there are nested terms, as an aid to keeping track of which go with which and the order in which they’re used is irrelevant…the problem could have been written: 8 + (−2[−3 − 2])² or any other combination.
Simplify (2 + 3²)²

Here’s a case where you have to take the exponent inside the parentheses first, simplify, then take the exponent outside the parentheses:

(2 + 3²)² = (2 + 9)² = 121

Here are some more examples:

Simplify 4( −2/3 + 4/3 )

Remember to simplify inside the parentheses first:

Simplify 4 − 3[4 −2(6 − 3)] ÷ 2

Simplify from the inside out: first the parentheses, then the brackets.  Also, be careful with taking the minus through the parentheses: remember that the −3 goes on everything inside the brackets and the −2 goes on everything inside the parentheses!

4 − 3[4 −2(6 − 3)] ÷ 2 =
=
=
=
=
=
4 − 3[4 − 2(3)] ÷ 2
4 − 3[4 − 6] ÷ 2
4 − 3[−2] ÷ 2
4 + 6 ÷ 2
4 + 3
7

Remember that, unless the grouping symbols tell you otherwise, division comes before addition, which is why this simplified in the end as 4 + 3, and not 10 ÷ 2.

Simplify 16 − 3(8 − 3)² ÷ 5

Remember to simplify inside the parentheses before you apply the exponent, because (8 − 3)² is not the same as 8² − 3².

16 − 3(8 − 3)² ÷ 5 =
=
=
=
=
16 − 3(5)² ÷ 5
16 − 3(25) ÷ 5
16 − 75 ÷ 5
16 − 15
1

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