**Learning Objectives**

After completing this tutorial, you should be able to:

- Evaluate an exponential expression.
- Simplify an expression using the order of operations.
- Evaluate an expression.
- Know when a number is solution to an equation or not.
- Translate an english expression into a math expression.
- 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. 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**

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.**

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:

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!

**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**

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.

**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.

***Exponent**

***Multiply**

***Add**

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.

***Exponent**

***Add in num. and subtract in
den.**

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.

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.

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

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

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.

Let’s step through some examples that help illustrate these ideas.

***Exponent**

***Multiply**

***Add**

***Subtract**

***Multiply**

***Add**

Two expressions set equal to each other.

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)

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.**

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

- read the problem carefully,
- pick out key words and phrases and determine their equivalent mathematical meaning,
- replace any unknowns with a variable, and
- put it all together in 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

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.

**Let’s see what we get for this answer:**

The product of 5 and a number

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 (=):**

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 =.

**Let’s put together everything going left to right:**

The quotient of 3 and a number is ½

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 =.

**Let’s 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.

Practice Problems 2a - 2b:Simplify each expression.

Practice Problem 3a:Evaluate the expression ifx= 1,y= 2, andz= 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. Letxrepresent the unknown number.

5a. 9 less than 5 times a number.

(answer/discussion to 5a)

(answer/discussion to 5a)

5b. The product of 12 and a number.

(answer/discussion to 5b)

(answer/discussion to 5b)

Practice Problems 6a - 6b:Write each sentence as an equation. Letxrepresent 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.