Onmatu- Pre-Algebra Article

Exponents-lesson


We start with the simplest example, that is when the exponent is 1. Base can be any natural number. By the way, do you know what natural numbers are?  Last time I've been in one city in Montenegro. On a bus station there I discovered they have platform  0. Is that natural to have? What do you think? 
Let's  start with our lesson. I wish you good luck in understanding how exponents work and I expect you write me your feedback about what do you need to better understand exponents.

 We denote by "^", "to the power", because of Word document we use.
Let's start.

2^1 = 2,

3^1 = 3,

4^1 = 4,..
and so on. We see that on the right hand side of these equations we have 1 base number (in our case 2, 3 and 4). Let's remember, on the left hand side of these equations we have exponent 1, and on the right hand side we have 1 base number.

After 1 there goes 2. Let's see what happens when we put exponent 2 to our base numbers 2, 3 and 4.

2^2 = 2 x 2 = 4,

3^2 = 3 x 3 = 9,

4^2 = 4 x 4 = 16.

We see that things changed. Can you tell what?

Hmmm, I see that on the left hand side of these equations we have exponent 2, and on the right hand side we have 2 base numbers and between them multiplication sign, x. Do you agree with me?

What do we remember here? Base number to the exponent 2 is the same as  (base number x base number). 

I bet you can do it yourself for exponent 3. Try.

From all of this we can conclude general rule for exponent e and base number b. That is

b^e = b x b x..x b  (on the right hand side we have b multiplied by itself e times). 

Let us see what happens if we put for our base number 0 instead of a natural number. Remember, our general rule applies here too. That is, we have

0^1 = 1,

0^2 = 0 x 0 = 0.

What is next? I am sure you know. 

0^3 = 0, right?

I think it is time for to conclude that for any exponent, e, we have

0^e = 0.

What is next? Of course, negative base. We start with -1. General rule still applies! :)

(-1)^1 = -1,

(-1)^2 = -1 x (-1) = 1,

(-1)^3 = -1 x (-1) x (-1) = -1,

(-1)^4 = 1.

What can we say here? First difference from the case when the base is a natural number is a BRACKET! Why do we put negative base into brackets? The reason is that (-1)^2  is different than -1^2. If we apply our general rule to -1^2, what do we get?

-1^2 = - (1 x 1) = -1.


Let us see few more examples to better understand the difference.

-1^3 = - (1 x 1 x 1) = -1,  (-1)^3 = -1 x (-1) x (-1) = -1, but

-1^4 = -(1 x 1 x 1 x 1) = -1, (-1)^4 = (-1) x (-1) x (-1) x (-1) = 1.

When an exponent is 0, we get  

1^0 = 1,

2^0 = 1,

3^0 = 1,

(-1)^0 = 1,

(-2)^0 = 1,


so we will remember for now  that every number to the power of 0 equals 1. At this stage we just understand power of 0 as a rule because we don't have enough of knowledge for that now. Sometimes in mathematics we have situations like this, and they are very good for developing our mental skills.  At one point, and it is a matter of our mental growth, this and similar things become very obvious. You don't know how that happened and that is a beauty of mathematics. We don't need to be discouraged when we don't understand something. It is so common in math.

 Since we are talking about rules that we will understand at one point, here is one more to remember,

0^0 is not defined!


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