Learning Objectives

1. Add and subtract real numbers.
2. Multiply and divide real numbers.
3. Evaluate a numeric exponential expression.
4. Evaluate a numeric radical expression.
5. Use the order of operations appropriately.

Introduction

Even in this day and age of calculators, it is very important to know these basic rules of operations on real numbers.  Even if you are using a calculator, you are the one that is putting the information into it, so you need to know things like when you are subtracting versus multiplying and the order that you need to put it in.  Also, if you are using a calculator you should have a rough idea as to what the answer should be. You never know, you may hit a wrong key and get a wrong answer (it happens to the best of us).  Also, your batteries in your calculator may run out and you may have to do a problem by hand (scary!!!).   You want to be prepared for those Murphy's Law moments.
 

We will start with the individual operations and then mix ‘em up using order of operations.

Tutorial

Step 2:   Attach their common sign to sum .
 
 
 

Example 1:   Find the sum -6 + (-8).

The sum of the absolute values would be 14 and their common sign is -.  That is how we get the answer of -14. 

You can also think of this as money - I know we can all relate to that.  Think of the negative as a loss.  In this example, you can think of it as having lost 6 dollars and then having lost another 8 dollars for a total loss of 14 dollars.

The sum of the absolute values would be 14.2 and their common sign is -. That is how we get the answer of -14.2.

You can also think of this as money - I know we can all relate to that.  Think of the negative as a loss.  In this example, you can think of it as having lost 5.5 dollars and then having lost another 8.7 dollars for a total loss of 14.2 dollars.

Step 2:   Attach the sign of the number that has the higher absolute value.

The difference between 8 and 6 is 2 and the sign of 8 (the larger absolute value) is -.  That is how we get the answer of -2. 

Thinking in terms of money: we lost 8 dollars and got back 6 dollars, so we are still in the hole 2 dollars.

*Take the difference of the numerators
and write over common denominator 6

*Reduce fraction
 

Thinking in terms of money: we had 2/3 of a dollar and lost 1/6 of a dollar, so we would come out ahead 1/2 of a dollar.

Note that if you need help on fractions, click on this link:  Fractions

Now, you do not have to write it out like this if you are already comfortable with it. This just gives you the thought behind it.

Subtracting 5 is the same as adding a -5. 

Once it is written as addition, I just follow the rules for addition, as shown above, to complete for an answer of -8.

Subtracting -5 is the same as adding 5. 

Once it is written as addition, I just follow the rules for addition, as shown above, to complete for an answer of 2.

Step 1:   Multiply or divide their absolute values. 
 

Step 2:   Put the correct sign.
 

The product of the absolute values 4 x 3 is 12 and they have opposite signs, so our answer is -12.

Note that if you need help on fractions, click on this link:  Fractions

The quotient of the absolute values 10/2 is 5 and they have the same signs, so our answer is 5.

*Mult. num. together
*Mult. den. together
*(+)(-) = -

*Reduce fraction

Note that if you need help on fractions, click on this link:  Fractions

Multiplying any expression by 0 results in an answer of 0.

Dividing 0 by any expression other than 0 results in an answer of 0.

Example 13:   Find the quotient 5/0.

Dividing by 0 results in an undefined answer.

The exponent tells you how many times a base appears in a PRODUCT.
 
 

Example 14:   Evaluate .
 

1. When there is no index number n, it is understood to be a 2 or square root.
2. When looking for the nth radical or nth root, you want the expression that when you raise it to the nth power you would get the radicand (what is inside the radical sign).

Note that we are only interested in the principal root and since 9 is positive and there is not a sign in front of the radical, our answer is positive 3.  If there had been a negative in front of the radical our answer would have been -3.

Now we are looking for the third or cube root of -1/8, which means we are looking for a number that when we cube it we get -1/8. 
Since -1/2 cubed is -1/8, our answer is going to be -1/2.

Please   Parenthesis or grouping symbols
Excuse   Exponents (and radicals)
My Dear   Multiplication/Division left to right
Aunt Sally  Addition/Subtraction left to right

When you do have more than one mathematical operation, you need to use the order of operations as listed above. You may have already heard of the saying "Please Excuse My Dear Aunt Sally".  It is just a way to help you remember the order you need to go in when applying the order of operations.
 

Example 19:    Simplify .

Using the Order of Operations we get:

*Multiply
*Add
 

Plugging in values for the variables and evaluating the expression using the Order of Operations we get:

*Exponent in num.

*Multiply in num. and den.

*Add in num. and subtract in den.
 

Practice Problems

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

Need Extra Help on These Topics?

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.