Polynomials Functions- Exponentials and Simplifying

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?

the you end up Each of those bunnies has a bunny, and you have eight bunnies and then 16 and 32 and 64 - your population just keeps growing.

so 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.

and 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 .

and we can write this When x =2, we can write this as x x x x x .

and so on and so forth, as x ^ n .

Now, in x ^ n , x is the base, n is the exponent, and we call this x to the n th power.

so 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, so we have population of 2^5.

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.

for one type of power that we look at and care about a lot is polynomials .

So we care about x to the n th 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 .

so 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.

x ^0=1 .

It’s a little strange, but if you think about it, you have to start somewhere.

So what are the properties, well Addition

There is no property for addition; there’s nothing special.

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.

Now, for multiplication , we do have some properties, like x ^3 x ^2.

Well, x ^3 = x x x and x ^2 = x x .

and now we are multiply thesese together, it will equal x x x x x , which is x ^5.

So for multiplication, ( x ^3)( x ^2) = x ^5.

and You can generalize that to ( x ^ n )( x ^ m )= x ^( n + m ).

Going back to the case of 2, we have 2^3 2^2 = 8 4 = 32 = 2^5.

What about division ?

for division, if I have 1 / ( x ^2), I can write that as x ^-2.

now This one’s a little but funky, but it’s a useful notation.

and You can use it in combination with multiplication to find things like (2^3) / (2^2).

well If you go ahead and solve this out, you find out that (2 ^ 8) and 2 ^ 4.

but i could say that (2^2) / (2^2) is 2^-2.

because 1 / 2^ is 2^-2

Then, I can use my multiplication property and say this is equal to 2^(3 - 2), where I’ve added my exponents of 3 and -2.

and, 2^(3 - 2) = 2^1 = 8 / 4 = 2.

the last property is that of a power (2^2)^3.

now This is that same as saying (2 2)(2 2)(2 2), which is 2^6.

so It’s reasonable to think that (2^2)^3 is the same as saying 2^(2 3), which is equal to 2^6.

and Again, you can generalize that by saying ( x ^ n )^ m = x ^( n m ).

so to summarize, here we looked at repeated multiplications, or our bunny problem, we wrote those as x ^ n , where x is the base and n is the exponent, which is called x to the n th power.

and we know for these that there are no addition rules, but there are multiplication , division and power rules.

There’s also that funny property where x ^0=1 .