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
xn
X is the base and n is the exponent.
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 x2, which is x* x, or x3, which is x * x * x. In general, we care about xn. Now let's look at some properties of xn. We know that x1=x, but what about x0? Well, x0 is NOT equal to zero. Instead, x0=1. It's a little strange, but if you think about it, you have to start somewhere.
x1 = x
x0 = not 0
Properties of x to the n
Properties of Polynomials
So what are the properties?
Addition Property
There are no addition properties; there's nothing special for addition. For example 23 + 2 2 does NOT equal 25. You can see this because 23 = 8 and 22 = 4 while 25 = 32, and 8 + 4 = 12, not 32.
Multiplication Property
Now, for multiplication, we do have some properties, like x^3 * x^2. Well, x^3 = x * x * x and x2 = x * x. So we know that when we multiply those together, it will equal x * x * x * x * x, which is x^5. So for multiplication, (x 3)(x 2) = x 5. You can generalize that to (x n)(xm)=x(n + m). Going back to the case of 2, we have 2^3 * 2^2 = 8 * 4 = 32 = 2 5.
Division Property
What about division? Well, if I have 1 / (x 2), I can write that as x -2. This one's a little but funky, but it's a useful notation. You can use it in combination with multiplication to find things like (2 3) / (2 2). If you solve this out, you have 8 / 4. We can also think of it as (2^3)(2^-2), because 1 / (2^2) 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. So, 2 (3 - 2) = 2 1 = 8 / 4 = 2.
Powers Property
Our last property is that of a power. Say we have (2 2) 3. This is the same as saying (2 * 2)(2 * 2)(2 * 2), which is 2^6. It's reasonable to think that (2 2)3 is the same as saying 2 (2 * 3), which is equal to 2 6. Again, you can generalize that by saying (x n)m = x(n*m).
Power
(22)3
This is the same as saying (2x2) (2x2) (2x2).
Lesson Summary
We looked at repeated multiplications, or our bunny problem, and we can write those as xn, where x is the base and n is the exponent, which is called x to the nth power. We know for these that there are no addition rules, but there are multiplication, divisionand power rules. There's also that funny property where x0=1.