**1. Introduction to Exponents**

An exponent is an elegant shorthand for multiplication.

Instead of 5 * 5 * 5, you can write 5

^{3}

Instead of 3 * 3 * 3 * 3 * 3 * 3 * 3, you can write 3

^{7}

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 4

^{2}= 4 * 4 = 16

And 4

^{3}= 4 * 4 * 4 = 64

And 4

^{1}= 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:

1

^{x}= 1

Using x-and-y notation, we can create a definition for the positive exponents.

Definition 1: Positive Exponents

xmeans^{y}xmultiplied with itselfytimes.xis called thebaseyis called thepower

**3. Multiplication of Exponents**

Multiplying exponents of the same base can be determined based on the above definition.

4

^{2}* 4

^{3}=

= (4 * 4) * (4 * 4 * 4)

= 4 * 4 * 4 * 4 * 4

= 4

^{5}

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:

5

^{5}* 5

^{10}= 5

^{15}

2

^{10}* 2

^{1000}= 2

^{1010}

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: x

^{y}* x

^{z}= x

^{(y+z)}

^{}

(1) We know that xQED 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.^{y}= x multipled to itself y times and that x^{z}= x multipled to itself z times. (Definition of Positive Exponents).

(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.

QED

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.

4

^{2}/ 4

^{1}=

= ( 4 * 4 ) / ( 4 ) =

= 16 / 4 = 4

= 4

^{1}

To divide two exponents of the same base, you simply subtract the two powers.

Here are some examples:

5

^{3}/ 5

^{1}= 5

^{2}4

^{10}/ 4

^{5}= 4

^{5}

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.

5

^{1}/ 5

^{3}= 1 / 5

^{2}

4

^{5}/ 4

^{10}= 1 / 5

^{5}

This leads us to a third definition:

Definition 3: Negative Exponents

xHere are some examples.^{(-y)}means that we have a fraction of 1 over x multiplied by itself y times.

5

^{-1}= 1 / 5

4

^{-3}= 1 / 4

^{3}

And what happens if the subtraction results in 0?

We can answer this with the following theorem:

**Theorem 2: x**

^{0}= 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.x^{0}= x^{(1 - 1)}

(2) Since 1 - 1 = 1 + (-1), we can rewrite this as:x^{(1 + -1)}

(3) Now x^{(1 + -1)}= x^{1}* x^{(-1)}by Theorem 1.

(4) Now,x, by Definition 3.^{(-1)}= 1/x

(5) So, we are left withx * (1/x) = 1

QED

Corollary 2.1: x

^{0}= 1 implies that x ≠ 0

(1) NowAnother way of saying this result is that 0x^{0}= x^{(1 - 1)}

(2) Which means thatx^{0}= x / x

(3) But this implies that x ≠ 0 since division by 0 is not allowed.

QED

^{0}just like 0/0 or even 1/0 is undefined.

We can summarize division of exponents with the following theorem.

**Theorem 3: x**

^{y}/ x^{z}= x^{(y - z)}Case I:y = z

In this casex.^{y}/ x^{z}= 1 = x^{0}= x^{(y - z)}

Case II:y > z

In division, we are able to cancel out all the common factors. Sincey > z, we cancel out z factors from both dividend and divisor and we are left withx.^{(y-z)}

Case III:y < z

Again, we cancel out common factors. Sincez > y, we are left with a fraction of1 / [xwhich, by definition 3, equals^{(z-y)}]x^{(-(z-y))}= x^{(y-z)}

QED

**5. Fractional Exponents**

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 x

^{1/2}* x

^{1/2}= x

^{(1/2 + 1/2)}by Theorem 1.

Now x

^{(1/2 + 1/2)}= x

^{(1)}= x.

So x

^{1/2}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: x**

^{1/n}= the nth root of x(1) x^{1/n}multiplied by itself n times equals x^{1/n + 1/n + 1/n + etc.}.

(2) Now 1/n + 1/n + etc. n times equals n/n which equals 1.

(3) Therefore x^{1/n}multipled by itself n times equals x^{1}

(4) And this is the very definition of an nth root.

QED

## 4 comments :

Excellent

indeed

excellent!

Four to the fifth power divided by four to the tenth power is one over four, (with the four to the fifth power)- right?

Four to the fifth power = 4^5 = 4*4*4*4*4

Four to the tenth power = 4^10 = 4*4*4*4*4*4*4*4*4*4

Four to the fifth power divided by four to the tenth power = 4^5/4^10 = 4^(5-10) = 4^(-5) = 1/(4^5) = 1/(4*4*4*4*4)

Four to the fifth power divided by four to the sixth power = 4^5/4^6 = 4^(5-6) = 4^(-1) = 1/(4^1) = 1/4.

I hope that helps.

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