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SIMPLIFYING EXPONENTIAL EXPRESSIONS VARIABLE IN THE EXPONENT
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Beginning Algebra
Tutorial 4:
Introduction to Variable Expressions and Equations


Learning Objectives

 
After completing this tutorial, you should be able to:
  1. Evaluate an exponential expression.
  2. Simplify an expression using the order of operations.
  3. Evaluate an expression.
  4. Know when a number is solution to an equation or not.
  5. Translate an english expression into a math expression.
  6. Translate an english statement in to a math equation.

 
 
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,  I know you will do a lot better than Baloo in this cartoon link.  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


 

Exponential Notation

 
An exponent tells you how many times that you write a base in a PRODUCT.

In other words, exponents are another way to write MULTIPLICATION.

Lets 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.  Lets 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: 


 
 
 
Example 1:   Evaluate 

 
In this problem, what is the base?

If you said 5, you are correct!

What is the exponent?

If you said 4, you are right!
 

Lets rewrite it as multiplication and see what we get for an answer:


 

*Rewrite the base 5, four times in a product
*Multiply

 
 
 
 
Example 2:   Evaluate 

 
In this problem, what is the base?

If you said 7, you are correct!

What is the exponent?

If you said 1, you are right!

Lets rewrite it as multiplication and see what we get for an answer:


 
*Rewrite the base 7, one time in a product

 
 
 
Example 3:   Evaluate 

 
In this problem, what is the base?

If you said 1/3, you are correct!

What is the exponent?

If you said 2, you are right!

Lets rewrite it as multiplication and see what we get for an answer:


 
*Rewrite the base 1/3, two times in a product
 

*Multiply
 


 
Note that when you have a 2 as an exponent,  which is also known as squaring the base.  In this problem we could say that we are looking for 1/3 squared.


 

Order of Operations

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 4:   Simplify .

 

*Multiply
*Add
*Subtract

 
 
Example 5:   Simplify 

 

*Inside (  )

*Exponent
*Multiply
*Add


 
 
 
Example 6:   Simplify .

 
Note that the absolute value symbol |   | is a fancy grouping symbol.  In terms of the order of operations, it would be including on the first line with parenthesis.

So in this problem, the first thing we need to do is work the inside of the absolute value.  And then go from there.


 
*Inside |    |
 
 

*Exponent
 

*Add in num. and subtract in den.
 


 
 
 
Variable

 
A variable is a letter that represents a number.

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.


 
 
Algebraic Expressions

 
An algebraic expression is a number, variable or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots.

2x + y, a/5, and 10 - r are all examples of algebraic expressions.


 

Evaluating an Expression

 
You evaluate an expression by replacing the variable with the given number and performing the indicated operation.

 
 
Value of an Expression

 
When you are asked to find the value of an expression, that means you are looking for the result that you get when you evaluate the expression.

 
So keep in mind that vary means to change - a variable allows an expression to take on different values, depending on the situation

For 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.

Lets step through some examples that help illustrate these ideas.


 
 
Example 7:   Evaluate the expression   when x = 4, y = 6, z = 8.

 
Plugging in the corresponding value for each variable and then evaluating the expression we get:

 

*Plug in 4 for x, 6 for y, and 8 for z

*Exponent
*Multiply
*Add
*Subtract


 
 
Example 8:   Evaluate the expression   when x = 3, y = 5, and z = 7.

 
Plugging in the corresponding value for each variable and then evaluating the expression we get:

 
*Plug in 3 for x, 5 for y, and 7 for z
*Exponent
 
 

*Multiply

*Add
 


 
 
Equation

Two expressions set equal to each other.


 

Solution

A 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)


 
Solution Set

Set of all solutions.


 
Example 9:   Is 2 a solution of ?

 
 
Replacing x with 2 we get:

 

*Plug in 2 for x
*Evaluate both sides

 
Is 2 a solution?

Since we got a TRUE statement (7 does in fact equal 7), then 2 is a solution to this equation.


 
 
 
Example 10:   Is 5 a solution of ?

 
 
Replacing x with 5 we get:

 
*Plug in 5 for x
*Evaluate both sides

 
Is 5 a solution?

Since we got a FALSE statement (16 does not equal 14), then 5 is not a solution.


 
 
 
Translating an 
English Phrase Into an 
Algebraic Expression

 
 
Sometimes, you find yourself having to write out your own algebraic expression based on the wording of a problem. 

In that situation, you want to 

  1. read the problem carefully,
  2. pick out key words and phrases and determine their equivalent mathematical meaning,
  3. replace any unknowns with a variable, and
  4. put it all together in an algebraic expression.
The following are some key words and phrases and their translations:

 
Addition: sum, plus, add to, more than, increased by, total 

 
Subtraction:  difference of, minus, subtracted from, less than, decreased by, less

 
Multiplication:  product, times, multiply, twice, of 

 
Division:  quotient divide, into, ratio 

 
 
 
Example 11:    Write the phrase as an algebraic expression.

  The sum of a number and 10.


 
In this example, we are not evaluating an expression, so we will not be coming up with a value.  However, we are wanting to rewrite it as an algebraic expression.

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 sum of a number and 10


*'sum' = +
*'a number' = variable x

 
 
Example 12:    Write the phrase as an algebraic expression.

The product of 5 and a number.


 
Again, we are wanting to rewrite this as an algebraic expression, not evaluate it.

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.
 

Lets see what we get for this answer:


 
The product of 5 and a number


*'product' = multiplication
*'a number' = variable x

 
 
 
Translating a Sentence into an Equation

 
Since an equation is two expressions set equal to each other, we will be using the same mathematical translations we did above.  The difference is we will have an equal sign between the two expressions.

The following are some key words and phrases that translate into an equal sign (=):


 
Equal Sign (=) :  equals, gives, is, yields, amounts to, is the same as

 
 
 
Example 13:    Write the sentence as an equation.  Let x represent the unknown number. 

The quotient of 3 and a number is ½.


 
Do you remember what quotient translates into?   If you said division, you are doing great.

'Is' will be replaced by the symbol =.

Lets put together everything going left to right:


 
The quotient of 3 and a number is ½


 
 
 
Example 14:    Write the sentence as an equation.   Let x represent the unknown number.

7 less than 3 times a number is the same as 0.


 
Do you remember what less than translates into?   If you said subtraction, you are doing great.

Do you remember what times translates into?   If you said multiplication, you are correct.

'Is the same as' will be  replaced by the symbol =.

Lets put together everything going left to right:


 
7 less than 3 times a number is the same as 0.


 
 
 
 
Practice Problems

 
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

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.


 
1a. 
(answer/discussion to 1a)
1b. 
(answer/discussion to 1b)

 


 

Practice Problems 2a - 2b: 

Simplify each expression.


 
2a. 
(answer/discussion to 2a)
2b. 
(answer/discussion to 2b)

 


 

Practice Problem 3a: 

Evaluate the expression if x = 1, y = 2, and z = 3.


 
3a. 
(answer/discussion to 3a)


 

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.


 
5a.   9 less than 5 times a number.
(answer/discussion to 5a)

 
5b.  The product of 12 and a number.
(answer/discussion to 5b)

 


 

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?

 
 
The following are web pages that can assist you in the topics that were covered on this page:
 

This webpage goes over the order of operations.


This webpage helps with translating english into math.


 

for some more suggestions.


 


 




All contents
Jan. 8, 2002