PEMDAS Review If We're Asked to Do Something Like 3 + 4 · 2, The

PEMDAS Review

If we’re asked to do something like 3 + 4 · 2, the question of “How do I go about doing this” would probably arise since there are two operations. We can do this in two ways: Choice Operation Description First 3+4 · 2=7 · 2=14 Add 3 and 4 and then multiply by 2 Second 3+4 · 2=3+8=11 Multiply 4 and 2 together and then add 3 It seems as though the answer depends on which how you look at the problem. But we can’t have this kind of flexibility in mathematics. It won’t work if you can’t be sure of a process to find a solution, or if the exact same problem can calculate to two or more differing solutions. To eliminate this confusion, we have some rules set up to follow. These were established at least as far back as the 1500s and are called the “order of operations.” The “operations” are addition, subtraction, multiplication, division, exponentiation, and grouping. The “order” of these operations states which operations take precedence over other operations.

We follow the following logical acronym when performing any arithmetic:

PEMD−−→−→AS

Step Operations Description Perform... 1 P Parenthesis all operations in parenthesis 2 E Exponentiation any exponents 3 −−→MD Multiplication and Division multiplication and division from left to right 3 −→AS Addition and Subtraction addition and subtraction from left to right Note 1. Multiplication and division can be in any order; that is, you perform multiplication or division from the left and ﬁnd the next multiplication or division and so forth. Similarly, this also applies to addition and subtraction. Note 2. PEMDAS is what we call a nested rule in parenthesis, that is, whenever you perform −−→−→ operations inside of parenthesis, a new PEMDAS is created for use inside of that parenthesis. Once −−→−→ all of the operations inside of a parenthesis are exhausted, you go back to your original PEMDAS. −−→−→ Depdnding on the number of levels of parenthesis, you could have many diﬀerent nested PEMDAS. −−→−→

So from the above example 3 + 4 · 2, we have the operations of addition and multiplication. Multiplication (Step 3) appears before addition (step 4), so we do it ﬁrst yielding 3+4 · 2 = 3 + 8. Now all we have left is addition so we perform that operation last giving us3+8=11. So3+4 · 2 = 3 + 8 = 11 is the correct solution. That is the second choice.

Now let’s try another problem. At ﬁrst glance, it may look complicated, but break it down into steps using the same logic in the example above. Suppose we wish to simplify 16 − 3(8 − 3)2 ÷ 5. We have the following operations here: subtraction, multiplication, parenthesis, exponentiation, and division. One thing to recall is that (8 − 3)2 =86 2 − 32. Operation Description 16 − 3(8 − 3)2 ÷ 5 Original problem

= 16 − 3(5)2 ÷ 5 Performed inside of parenthesis ﬁrst (Step 1)

= 16 − 3 · 25 ÷ 5 Performed exponentiation next (Step 2)

= 16 − 75 ÷ 5 Performed multiplication (Step 3)

= 16 − 15 Performed division (Step 3)

= 1 Performed subtraction (Step 4) P One way to remember PEMD−−→−→AS is to create a mnemonic for it. A widely used one is lease Excuse My Dear Aunt Sally.

Now let’s try one with variables. Suppose we wish to simplify 14x + 5[6 − (2x + 3)]. Don’t let the variables muddle up your process, you still have the same order of operations. Here the operations are: addition, parenthesis, subtraction, and multiplication.

Operation Description 14x + 5[6 − (2x + 3)] Original problem

= 14x + 5[6 − 1(2x + 3)] Rewrite of the problem. Recall that you can put a 1 in front of parenthesis if no coeﬃcient already exists.

= 14x + 5[6 − 2x − 3] We must perform all operations inside of the parenthesis ﬁrst (Step 1). Within the parenthesis, we have a nesting so we start. x with a new PEMD−−→−→AS. Recall that 2 + 3 cannot be combined since one is a variable and one is a constant. This is because they are not like terms. So we must distribute the −1 by multiplying through (Step 3).

= 14x + 5[3 − 2x] Performed subtraction of like terms (Step 4). Now the inside of the parenthesis is ﬁnished. We resume our original PEMD−−→−→AS.

= 14x + 15 − 10x Distribution of 5 by multiplying through (Step 3).

= 4x + 15 Performed subtraction of like terms (Step 4).

Remember, as long as you have do perform any arithmetic, PEMD−−→−→AS will guide you. It is as important as knowing the diﬀerence between a noun and a verb, so make sure to remember it!