Learning Objectives
Introduction
This tutorial will go over some key definitions and phrases used when specifically working with algebraic expressions as well as evaluating them. We will also touch on the order of operations. It is very IMPORTANT that you understand some of the math lingo that is used in an algebra class, otherwise it may all seem Greek to you. Knowing the terms and concepts on this page will definitely help you build an understanding of what a variable is and get you more comfortable working with them. Variables are a HUGE part of algebra, so it is very important for you to feel at ease around them in order to be successful in algebra. So let's get going and help you get on the road to being variable savvy.
Tutorial
In other words, exponents are another way to write MULTIPLICATION.
Let’s illustrate this concept by rewriting the product (4)(4)(4) using exponential notation:
In this example, 4 represents the base and 3 is the exponent. Since 4 was written three times in a product, then our exponent is 3. We always write our exponent as a smaller script found at the top right corner of the base.
You can apply this idea in the other direction. Let’s say you have it written in exponential notation and you need to evaluate it. The exponent will tell you how many times you write the base out in a product. For example if you had 7 as your base and 2 as your exponent and you wanted to evaluate out you could write it out like this:
If you said 5, you are correct!
What is the exponent?
If you said 4, you are right!
Let’s rewrite it as multiplication and see what we get for an answer:
If you said 7, you are correct!
What is the exponent?
If you said 1, you are right!
Let’s rewrite it as multiplication and see what we get for an answer:
If you said 1/3, you are correct!
What is the exponent?
If you said 2, you are right!
Let’s rewrite it as multiplication and see what we get for an answer:
*Multiply
Please Parenthesis or grouping symbols
Excuse Exponents (and radicals)
My Dear Multiplication/Division
left to right
Aunt Sally Addition/Subtraction
left to right
*Exponent
*Multiply
*Add
So in this problem, the first thing we need to do is work the inside of the absolute value. And then go from there.
*Exponent
*Add in num. and subtract in
den.
Don't let the fact that it is a letter throw you. Since it represents a number, you treat it just like you do a number when you do various mathematical operations involving variables.
x is a very common variable that is used in algebra, but you can use any letter (a, b, c, d, ....) to be a variable.
2x + y, a/5, and 10 - r are all examples of algebraic expressions.
Evaluating an ExpressionFor example, the area of a rectangle is length times width. Well, not every rectangle is going to have the same length and width, so we can use an algebraic expression with variables to represent the area and then plug in the appropriate numbers to evaluate it. So if we let the length be the variable l and width be w, we can use the expression lw. If a given rectangle has a length of 4 and width of 3, we would evaluate the expression by replacing l with 4 and w with 3 and multiplying to get a value of 4 times 3 or 12.
Let’s step through some examples that help illustrate these ideas.
*Exponent
*Multiply
*Add
*Subtract
*Multiply
*Add
Two expressions set equal to each other.
SolutionA value, such that, when you replace the variable with
it,
it makes
the equation true.
(the left side comes out equal to the right side)
Set of all solutions.
Since we got a TRUE statement (7 does in fact equal 7), then 2 is a solution to this equation.
Since we got a FALSE statement (16 does not equal 14), then 5 is not a solution.
In that situation, you want to
The sum of a number and 10.
It looks like the only reference to a mathematical operation is the word sum. So, what operation will we have in this expression?
If you said addition, you are correct!!!
The phrase 'a number' indicates that it is an unknown number. There was no specific value given to it. So we will replace the phrase 'a number' with the variable x. We want to let our variable represent any number that is unknown
Putting everything together, we can translate the given english phrase with the following algebraic expression:
The product of 5 and a number.
This time, the phrase that correlates with our operation is 'product' - so what operation will we be doing this time? If you said multiplication, you are right on.
Again, we have the phrase 'a number', which again is
going to be replaced
with a variable, since we do not know what the number is.
Let’s see what we get for this answer:
The following are some key words and phrases that translate into an equal sign (=):
The quotient of 3 and a number is ½.
'Is' will be replaced by the symbol =.
Let’s put together everything going left to right:
7 less than 3 times a number is the same as 0.
Do you remember what times translates into? If you said multiplication, you are correct.
'Is the same as' will be replaced by the symbol =.
Let’s put together everything going left to right:
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.
Practice Problems 1a - 1b: Evaluate.
Practice Problems 2a - 2b: Simplify each expression.
Practice Problem 3a: Evaluate the expression if x = 1, y = 2, and z = 3.
Practice Problems 4a - 4b: Decide whether the given number is a solution of the given equation.
4a. Is 0 a solution to ?
(answer/discussion
to 4a)
4b. Is 8 a solution to ?
(answer/discussion
to 4b)
Practice Problems 5a - 5b: Write each phrase as an algebraic expression. Let x represent the unknown number.
Practice Problems 6a - 6b: Write each sentence as an equation. Let x represent the unknown number.
6a. The sum of 10 and 4 times a number is the
same as 18.
(answer/discussion
to 6a)
6b. The quotient of a number and 9 is 1/3.
(answer/discussion
to 6b)
Need Extra Help on these Topics?
http://www.sosmath.com/algebra/fraction/frac3/frac39/frac39.html
This webpage goes over the order of operations.
http://www.purplemath.com/modules/translat.htm
This webpage helps with translating english into math.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 42, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.