WebQuest: Properties of Exponents

Why are we doing this?

Being able to simplify exponential expressions is a crucial step in understanding basic algebraic operations. The properties covered in this section need to be understood, not memorized. If you've ever say to yourself "I can't remember if I'm supposed to add these exponents or multiply them" then you are trying to memorize them. And, just like most things you memorize, you will be destined to forget them.

Technical Stuff:

Understanding these properties is very easy - as long as we think about it. We already know exponents imply repeated multiplication. We will use that fact to help us understand the following properties. xa · xb = xa+b This is called the product rule. In English, this says: To multiply two exponential expressions that have the same base, keep the base and add the exponents. Why does this work? Consider what it means to be an exponent. x3 means x · x · x, and x5 means x · x · x · x · x. So, if we multiply x3 · x5 we literally have (x · x · x) · (x · x · x · x · x). 3 x's from the first expression and 5 X's from the second one gives a total of 8 X's when they are put together. Thus, x3 · x5 = x3+5= x8. xa/xb = xa-b This is called the quotient rule. In English, this says: To divide two exponential expressions that have the same base, keep the base and subtract the exponent in the denominator from the exponent in the numerator. Why does this work? Once again, consider what it means to be an exponent. x5 means x · x · x · x · x, and x3 means x · x · x. So, if we divide x5/x3 we literally have (x · x · x · x · x)/(x · x · x). Upon removing 3 X's from the both the numerator and denominator, we are left with a total of 2 X's in the numerator. Thus, x5/x3 = x5-3= x2. Example: Simplify the following. (xa)b = xab This one is called the power rule. In English, this says: To raise a power to a power, keep the same base and multiply the exponents. Why does it work? Consider what it means to be an exponent (do you see a common theme here?). x3 means x · x · x, and y5 means y · y · y · y · y. So, if we multiply (x3)5 we literally have (x · x · x) · (x · x · x) · (x · x · x) · (x · x · x) · (x · x · x). 3 X's repeated 5 times is 15 X's, and 3 × 5 = 15. Thus, (x3)5 = x3 · 5= x15. This rule and the product rule are very similar and, hence, very easy to confuse. One way to keep them straight is to notice that, when the common bases each have their own exponent in a multiplication problem, we will add the exponents. But if we see a single base that looks like it has two exponents, we will multiply those exponents. (xy)a = xaya and (x/y)a = xa/ya These two are variations of the power rule, and are a consequence of what it means to be an exponent. In fact, when ever we are unsure how to proceed, we can always fall back on the meaning of an exponent. If we have to, we write out as many steps as we need in order to see what happens.

First, let's do some math.
	Simplify the following expressions. a. b. Answer --> [x^7]/[5] Answer --> [64]/[w^15]     c. d. (x3)3 Answer --> a^30b^12 Answer --> x^9     e. x3·x3 f. x3 + x3 Answer --> x^6 Answer --> 2x^3

Now, let's explore a website.
	Go to the Practice with Exponents website and experiment with several problems at each level.

Questions to think about.

1. Under what conditions does xa represent a negative number? Why?   Answer --> x < 0, and a is an odd number.     2. Using the quotient rule, explain why 50 = 1.   Answer --> 1 = 5^2/5^2 = 5^(2-2) = 5^0     3. To simplify (x4x2)3, do you use the product rule or the power rule first?   Answer --> It doesn't matter.     4. Suppose the side length of a cube is three times the side length of a second cube. How do the volumes compare?   Answer --> The volume of the first cube is 27 times larger.