# Category Archives: Exponents

## Exponents

### Exponents

**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 thebase

yis 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 x

^{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

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.

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.

a is refered to as the quotient.

b is refered to as the dividend.

c is refered to as the divisor.

Division with exponents of the same base can also be determined based on the definition for positive exponents:

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 thedivisor 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

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 / 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 that

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) = 1QED

We can also introduce a corollary to this theorem. A corollary is a small proof that is derived directly from the logic of a theorem.

Corollary 2.1: x^{0} = 1 implies that x ≠ 0

(1) Now

x^{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

Another way of saying this result is that 0^{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 = zIn this case

x.^{y}/ x^{z}= 1 = x^{0}= x^{(y – z)}Case II:

y > zIn 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 withx.^{(y-z)}Case III:

y < z

Again, we cancel out common factors. Sincez > y, we are left with a fraction of

1 / [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