An exponent is an elegant shorthand for multiplication.
Instead of 5 * 5 * 5, you can write 53
Instead of 3 * 3 * 3 * 3 * 3 * 3 * 3, you can write 37
The number that gets multiplied is called the base. The number of multiplications that occur is called the power. So, in the above example, 3 is the base and 7 is the power.
Of course, this method only applies when the power is a positive integer. Later on, I will discuss what it means when a power is 0, positive, or even a fraction.
So 42 = 4 * 4 = 16
And 43 = 4 * 4 * 4 = 64
And 41 = 4 = 4
2. x and y notation
In mathematics, when we want to talk about "any", we use a letter such as x or y or z. For example, if we wanted to say that 1 to any power equals 1, we could write this as follows:
1x = 1
Using x-and-y notation, we can create a definition for the positive exponents.
Definition 1: Positive Exponents
xy means x multiplied with itself y times.
x is called the base
y is called the power
3. Multiplication of Exponents
Multiplying exponents of the same base can be determined based on the above definition.
42 * 43 =
= (4 * 4) * (4 * 4 * 4)
= 4 * 4 * 4 * 4 * 4
So, when exponents get multiplied, if they have the same base, you can add the powers and create a new exponent.
Here are some more examples:
55 * 510 = 515
210 * 21000 = 21010
Of course, this does not work if two exponents have a different base.
In mathematics, a method such as this can be presented as a theorem. A theorem is any statement that can be derived from previous results.
In this case, we are able to prove a theorem regarding the method of adding the powers of the same base. Here's the theorem
Theorem 1: xy * xz = x(y+z)
(1) We know that xy = x multipled to itself y times and that xz = x multipled to itself z times. (Definition of Positive Exponents).QED is put at the end of a proof to show it is done. It is an abbreviation for a latin phrase that means basically that the proof is finished. It serves the same purpose in a proof as a period does in a sentence.
(2) Multiplying all those x's, we have (y + z) x's multiplied together.
(3) Now x multiplied to itself (x + z) times = x(y + z) by the Definition of Positive Exponents.
4. Division of Exponents
To talk about division, it is useful to introduce the following definition:
Definition 2: Division
a = b / c means a is equal to b divided by c.Division with exponents of the same base can also be determined based on the definition for positive exponents:
a is refered to as the quotient.
b is refered to as the dividend.
c is refered to as the divisor.
42 / 41 =
= ( 4 * 4 ) / ( 4 ) =
= 16 / 4 = 4
To divide two exponents of the same base, you simply subtract the two powers.
Here are some examples:
53 / 51 = 52
410 / 45 = 45
Now, what happens if we are dividing by a number greater than the top (in other words, where the divisor is greater than the dividend)? In this case, we are left with a fraction.
51 / 53 = 1 / 52
45 / 410 = 1 / 55
This leads us to a third definition:
Definition 3: Negative Exponents
x(-y) means that we have a fraction of 1 over x multiplied by itself y times.Here are some examples.
5-1 = 1 / 5
4-3 = 1 / 43
And what happens if the subtraction results in 0?
We can answer this with the following theorem:
Theorem 2: x0 = 1
(1) By basic arithemitic, we know thatWe can also introduce a corollary to this theorem. A corollary is a small proof that is derived directly from the logic of a theorem.
x0 = x(1 - 1)
(2) Since 1 - 1 = 1 + (-1), we can rewrite this as:
x(1 + -1)
(3) Now x(1 + -1) = x1 * x(-1) by Theorem 1.
(4) Now, x(-1) = 1/x, by Definition 3.
(5) So, we are left with x * (1/x) = 1
Corollary 2.1: x0 = 1 implies that x ≠ 0
(1) Now x0 = x(1 - 1)Another way of saying this result is that 00 just like 0/0 or even 1/0 is undefined.
(2) Which means that x0 = x / x
(3) But this implies that x ≠ 0 since division by 0 is not allowed.
We can summarize division of exponents with the following theorem.
Theorem 3: xy / xz = x(y - z)
Case I: y = z5. Fractional Exponents
In this case xy / xz = 1 = x0 = x(y - z).
Case II: y > z
In division, we are able to cancel out all the common factors. Since y > z, we cancel out z factors from both dividend and divisor and we are left with x(y-z).
Case III: y < z
Again, we cancel out common factors. Since z > y, we are left with a fraction of
1 / [x(z-y)] which, by definition 3, equals x(-(z-y)) = x(y-z)
There is more that we can talk about. What about fractional exponents such as x(1/2)?
It turns out that based on our definitions, corrolaries, and theorems, we are now ready to take on fractional exponent.
Let's start with 1/2.
We know that x1/2 * x1/2 = x(1/2 + 1/2) by Theorem 1.
Now x(1/2 + 1/2) = x(1) = x.
So x1/2 is none other than the square root of x.
Let's start out by looking at a definition for what a root is.
Definition 4: an nth root of x is a number that multiplied n times equals x.
Sometimes, nth roots are whole numbers. The cube root of 27 is 3 since 3 * 3 * 3 = 27.
Likewise, the 4th root of 16 is 2.
1 is its own 5th root since 1 * 1 * 1 * 1 * 1 = 1.
This gives us our last theorem:
Theorem 4: x1/n = the nth root of x
(1) x1/n multiplied by itself n times equals x1/n + 1/n + 1/n + etc..
(2) Now 1/n + 1/n + etc. n times equals n/n which equals 1.
(3) Therefore x1/n multipled by itself n times equals x1
(4) And this is the very definition of an nth root.