# Polynomials Functions: Exponentials and Simplifying

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• 0:34 ''x'' to the ''n''th Power
• 1:43 Powers in Daily Life
• 2:30 Polynomials
• 3:26 Properties of Polynomials
• 7:07 Lesson Summary

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Lesson Transcript
Instructor: Robert Egan
How do we keep track of a rapidly multiplying population of bunnies? Well, those are simply powers of 2. Review powers and simplify problems with exponents in this lesson.

## Bunnies Galore

Have you ever heard of that bunny problem? You know, the one where you start out with one bunny, and then all of a sudden you have two bunnies, and each of those bunnies has a bunny, so you end up with four bunnies? Each of those bunnies has a bunny, and you have eight bunnies and then 16 and 32 and 64 - your population just keeps growing!

## x to the nth Power

You could've said that your population started out with one times two bunnies times two bunnies times two bunnies and so on and so forth. If we ignore that one, because it doesn't really matter in this case, you end up with a case of repeated multiplication: 2 * 2 * 2 * 2 * 2 ... When x=2, we can write this as x * x * x * x * x ... and so on and so forth, which gives us x^n. Now, in x^n, x is the base, n is the exponent, and we call this x to the nth power. In the case of our population, we had 2 * 2 * 2 * 2 * 2, and let's just cap it off there. So we have five 2's for a population of 2^5.

## Powers in Daily Life

Now these powers are used all over in math and really all over the world. For example, if we want to look at mummies and know how old they are, we use an approach like carbon dating. And, carbon dating is used with powers, which might be something like 2.7^-t, where t is time. So we're using a power to determine the age of a mummy. Another example is the metric system. In the metric system, we're using powers that look like 10^x meters. Now, if x=-10, you're looking at something about the size of an atom. If x=20, you're looking at something roughly the size of the galaxy.

## Polynomials

One type of power that we look at and care about a lot is polynomials. So we care about x to the nth power, where n is some number, and we care about these because they are things like x or x^2, which is x * x, or x^3, which is x * x * x. In general, we care about x^n. Now let's look at some properties of x^n. We know that x^1=x, but what about x^0? Well, x^0 is NOT equal to zero. Instead, x^0=1. It's a little strange, but if you think about it, you have to start somewhere.

## Properties of Polynomials

So what are the properties?

There are no addition properties; there's nothing special for addition. For example 2^3 + 2^2 does NOT equal 2^5. You can see this because 2^3 = 8 and 2^2 = 4 while 2^5 = 32, and 8 + 4 = 12, not 32.

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