# Algebra: Exponents

Updated on March 5, 2014

## Explaining Exponents

In the years I've been tutoring, I have noticed that many people find the concepts of exponents to be a very daunting one. But exponents are really only an easier, more understandable way of expressing larger, more cumbersome numbers. I hope you feel less intimidating after reading this.

## Firstly, Some Simple Exponent Rules

Before you check out the rest of this lens, keep 2 things in mind:

1) You cannot add numbers with exponents UNLESS they have the same exponent and base AND

2) The base (that is the number being raised to the power) needs to be the same to be able to multiply or divide them.

For example, you cannot add the terms in figure 1 because they have different exponents. You cannot simplify the terms in figure 2 because they have different bases (a, b and c).

## Explaining Exponents

An exponent means that's how many times you multiply a number by itself. As you can see from the picture, A squared (that's A to the 2nd power) is A times A. A cubed (A to the 3rd power) is A times A times A. A to the 4th power is A times A times A times A.

## Multiplying Exponents

Multiplying numbers with exponents is relatively easy -- all you do is add the exponent.

If you look at the graphic, you can see why that works -- you are adding the "A"s in the multiplication problem. The "A" squared represents two "A"s multiplied by each other, the "A" cubed represents three "A"s multiplied by each other. So you add the "A"s and you get "A" to the fifth power.

## Dividing Exponents

Dividing exponents is pretty much just as easy as multiplying, but reversed. To divide numbers with exponents, you subtract the exponents. As you can see from this graphic, "A" cubed divided by "A" squared is "A" to the first, or "A".

## Raising an Exponent to Another Power

In order to raise an exponent to another power, you multiply the exponents. As you can see from the graphic, each "A" Squared appears three times -- so if you count the "A"s you will see there is a total of 6 "A"s.

## Negative Exponents

A negative exponent is just 1 over the same number exponent -- just as multiplying has you adding exponents, using negative exponents would have you add the exponents --

## Scientific Notation

Scientific Notation is the term for using the powers of 10 (each of which represent one zero) to express complex numbers. For example, in the graphic as the right, 406,000,000 is difficult to deal with. So if you express this as 4.06 X 10 to the 8th power, then it is easier to work with. On the other hand, .000000406 is 4.06 X 10 to the -7 (that is 1/10 to the seventh power).

10 to the 8th power is a 1 with 8 zeroes, 100,000,000 and 4.06 times that means that the 4 would be where the 1 is in 100,000,000 and the .06 would follow the 4 into the next two spots in the number.

## Math Themed Items from Compugraph Designs' Printfection Store

In addition to our Cafe Press and Zazzle sites (see modules above), we also have a store on "Printfection" which includes cutting boards (good wedding or housewarming gifts), mugs and cups, tees, etc.

This cutting board is only one of several Math themed items at our store:

Compugraphd Printfection site

(Click on the picture to go directly to this product's page)

## Compugraph Designs Art Now Site

"Art Now" is another "Print on Demand" site. They have a nice collection of clocks and watches, including the one pictured here (with math symbols on it). Click on the picture to see the entire site.

## Math Themed Items from Compugraph Designs' Printfection Store

In addition to our Cafe Press and Zazzle sites (see modules above), we also have a store on "Printfection" which includes cutting boards (good wedding or housewarming gifts), mugs and cups, tees, etc.

This mug is only one of several Math themed items at our store:

Compugraphd Printfection site

(Click on the picture to go directly to this product's page)

## Popular

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• ### Math Project Ideas: Examples of Project-Based Learning

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• AUTHOR

compugraphd

7 years ago

@MBCOnline: B"H

Well, if you ever need any more help, I also tutor -- if you're not in my geographical area, we could work something out with skype (and Paypal, perhaps?), if you'd like...

• MBCOnline

7 years ago

Thanks for making this so easy to understand! My first college algebra class starts in two weeks - I am very glad to have found you on Squidoo! No doubt I will be visiting more of your lenses!

• anonymous

8 years ago

"An exponent means that's how many times you multiply a number by itself." This statement is faulty.

A better wording is "an exponent tells you how many times the base is used in a multiplication" or "an exponent tells you how many times the base is being multiplied"

A number squared is not a number multiplied by itself 2 times, it is multiplied by itself once.

A number cubed is not a number multiplied by itself 3 times, it is multiplied by itself twice.

• kpp2385

8 years ago

these concepts are simple but the way you have explained them, it would be very easy for students to grasp them quickly. nice work

• Miha Gasper

8 years ago from Ljubljana, Slovenia, EU

Yes, you made clear presentations. But I am pretty familiar with exponents from my college days, so i can only say: keep up good work!

• AUTHOR

compugraphd

8 years ago

@anonymous: B"H

Please tell me where I said this. I said you can't multiply if the bases are different and that you can't ADD if the exponents are different.

• anonymous

8 years ago

Absolutely not. Yes, you can multiply exponential terms that have the same bases and different exponents. This explanation is incorrect.

• AUTHOR

compugraphd

8 years ago

@Blackspaniel1: ×"×

It can either be 0 or 1, depending on how you interpret it (and what it's "job" is in the equation). Since most students won't encounter 0**0, I'm not too worried. Check out this link -- http://mathforum.org/dr.math/faq/faq.0.to.0.power.... --

This sort of reminds me of repeating decimals. My brother (who is also a math geek) and I have had a lifelong disagreement about what the value of .9 repeating 9 equals -- he says it's as close to 1 as you can get without being 1, I say it IS 1 (since .1 repeating 1 equals 1/9, .2 repeating 2 equals 2/9, etc., .9 repeating 9 equals should equal 9/9 and that's 1).

• anonymous

8 years ago

I am sure this will benefit students!

• Blackspaniel1

9 years ago

Nice start. How about 0 to the 0 power?

• Eliza Rayner

9 years ago from Boulder, Colorado

Great and simple explainations, I can see how this is helpful to students, I will suggest it for my high school students if they need math help.

• Heidi Reina

9 years ago from USA

the simplest clearest explanations are always the best. Blessed by a Squidangel :)

• sorana lm

9 years ago

Very nice lens, clear, easy to follow and straight to the point.

• Joan Hall

10 years ago from Los Angeles

Hi! I've added this to the featured lenses on "Math Instruction Lenses" lensography.

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