Exponents and the Order of Operations

Exponents are a shorthand for writing a multiply that uses only one number several times.

5²  is 5 with an exponent of 2 means 5 times 5 = 25

6 with an exponent of 3 or 6³ means 6 times 6 times 6 = 216.   I wrote the 6 down three times.

But notice the differences:

-6² = -36, but (-6)²= +36 and (-6²)= -36 One of these is tricky!!

The parenthesis around the -6 tells me the -6 is squared (another word for exponent of 2).   We write the 6 down 2 times and then we multiply.

SEE:

(-6)²= (-6)(-6) = +36 .We write the -6 down two times and multiply.   Recall two negatives multiplied yields a positive answer.

Without the parenthesis only the 6 is squared. We write the 6 down two times and multiply.

- 6²= - (6)(6) = - 36.

With the parenthesis BUT the 2 is inside: (-6²) = - 36 This is really the same as -6². tricky one!!

Do you see that the exponent 2 is inside the parenthesis?

That means only the 6 is squared not the whole parenthesis, not the -6.

Thus the -6² and the (-6²) are really the same. They equal - 36.   
  And (- 6)² is   +36.

Write these down on a note card or place a sticky note. 

You will see this on the QUIZ below. .

We have studied the FOUR operations of math:

addition, subtraction, multiplication and division.

And, you have studied exponents.

You know how to do each of these, but can you do them if I mix 3 or 4 of them together?

Study these carefully.

A. -60 divided by 10 * 2

B. -60 divided by 5 * 4

C. 60 divided by (-5 * 4)

D. -60 divided by 2 divided by 5 * 2

Which of the above means - 3, and which means -12? 

  Which one has another number for its answer?

You must understand the above problems and these below about exponents before leaving this lesson.

Do not jump into these without studying the rules.

Most people remember this somewhat, but there are pitfalls that many always seem to fall into.

When the operations and exponents are MIXED together which do we do first?

THE ORDER is important. An agreement has been made.

All calculators and teachers and computers know it.

It is time for you to learn it too.

The Order of Operations AGREEMENT:

You must follow the order.

Step 1 requires you to SEARCH for the grouping symbols and then complete the work INSIDE them.

Grouping symbols are the parenthesis ( ), brackets[ ], bars| |, braces{ } and even the long fraction bar. While working "inside" you follow these 4 steps.

Step 2 requires you to SEARCH for the exponents in the expression and evaluate them.

Step 3 states that you are to do the multiply and the divides but not necessarily in that order. Many students think that you do the multiplies and then do the divides. That is WRONG.

You do NOT search for the multiplies and then search for the divides!!!!! They MUST be worked from left to right. Before beginning step 3 be sure you have returned to the left side of the problem. Then do the divide or multiply as you move across the problem going toward the right. If the divide is left most then divide first, but if the multiply is left most then you multiply first.

Step 4 states that you are to do the add and subtracts, but not necessarily in that order.

You do NOT search for the adds and then search for the subtracts!!!!! Many students think that you do the adds and then do the subtracts. That is WRONG. They MUST be worked from left to right. Before beginning step 4 be sure you have returned to the left side of the problem. Then do the subtracts or adds as you move across the problem going toward the right.

ORDER of OPERATIONS EXAMPLES

Example 1 : 
    - 3 + 36 ÷ (3)( 2) = There are no grouping symbols nor exponents.

Do we do the division or the multiply first? WRITE these on your paper going down the page as I have done.

You MUST learn to write them this way. Otherwise you will only be able to do the simple ones.

- 3 + 36 ÷ (3)( 2) =

-3 + 12( 2) =

-3 + 24=21

I started with step 3 since there are no grouping symbols and no exponents. Begin on the left  and do multiply or divide as you come to them. Many people will do the (3)( 2) before the divide; that is wrong.

But you must begin on the left. 

                              Multiply is done before division only if it is left most.

Then begin again on the left and do add or subtract as you come to them.

Try this problem:

Example 2:
    45 divided by (3)( 6 - 3) =

we have 45 ÷ (3)( 3)

We did the work inside the parenthesis. Now do we multiply the threes or divide the 45 by 3?

= 15(3)= 45.

Do not be tempted to multiply to get 9. That would mean that you did not return to the left to begin step 3.

Example 3 :  
       3 - ( 5 + 8*2÷4 - 1)

The (5+8*2÷4-1) is done first becauseof the parenthesis, but the Steps 2 through 4 must be followed

as you work inside the parenthesis. Step 2 (exponents, not needed). Step 3 says do multiply or divide ( start on the left).

So do 8*2 then divide by the 4. Then step 4 says do add or subtract( start on the left).

5+4 - 1=8. We have finished inside the parenthesis. 3-(5+8*2÷4 -1)= 3-(5+4 -1) =

I hope you are writing these on your paper as I have them typed. 3 - (8) = -5

Example 4: 
       5 + (- 6^2) NOTE: the ^2 means exponent of 2.   We use that on computers sometimes.

5 - 36 = -31 Do you know why this has - 36 and not +36? We discussed that above.

Sometimes we use the "/" to mean "divide by".

Example 5 : 
          60/3 + 2(-2) ^2 =     What is the first thing to do?

There is NO Step 1. We begin with Step 2, SEARCH for the exponents.

Then we go to Step 3, NO SEARCHING just start on the left.

We have one division and then one multiply for Step 3. And, we have one addition to do for Step 4.

60/3 + 2(-2)^2=            The ^2 means exponent of 2.    We use that on computers sometimes.

60/3 + 2(4) =

20 + 8 = 28

Example 6:
         35 - (- 4 -2)^2  - 8^2 = Step 1: Search for the grouping symbols and

work INSIDE the ( ).

35 - (- 4 -2)^2 - 8^2=

35 - (- 6)^2 - 8^2= Search for the exponents for Step 2 and evaluate those.

35 - (+36) - 64 = Now what? Step 3. START on the LEFT.

Begin on the left and do what? The subtract or the multiply or the add?

Well , of course we multiply before we subtract.     I hope you know -(+36) is same as -36.

35 - 36 - 64 =

-1 - 64.

The final answer is -65. Did you get that?

You really should copy some of the examples above into your notebook and use them to help you.

Order of Operations Game.      Try to get a score of 3000.