Answer:

Exponents

Let’s start with a REVIEW of the basics.

2

5 = 2 × 2 × 2 × 2 × 2

When writing 2

5

, the 2 is the base, and the 5 is the exponent or power.

We generally think of multiplication when we see a number with an exponent.

With 2

5

, we think of five twos multiplied together.

So let’s make a list to practice this way of thinking.

2

2 = 2 × 2 = 4

2

3 = 2 × 2 × 2 = 8

2

4 = 2 × 2 × 2 × 2 = 16

2

5 = 2 × 2 × 2 × 2 × 2 = 32

2

6 = 2 × 2 × 2 × 2 × 2 × 2 = 64

2

7 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

2

8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

2

9 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512

2

10 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

2

11 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2048

2

12 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 4096

2

13 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 8192

2

14 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 16,384

So, obviously, exponents are all about multiplication.Let’s reverse the order of our list.

2

14 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 16,384

2

13 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 8192

2

12 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 4096

2

11 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2048

2

10 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

2

9 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512

2

8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

2

7 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

2

6 = 2 × 2 × 2 × 2 × 2 × 2 = 64

2

5 = 2 × 2 × 2 × 2 × 2 = 32

2

4 = 2 × 2 × 2 × 2 = 16

2

3 = 2 × 2 × 2 = 8

2

2 = 2 × 2 = 4

Now, take a LOOK at the products from the top of the list to the bottom. Instead of

multiplying by two, we are dividing by 2 to get the next one.

So, let’s continue the list by dividing by 2.

2

1 = 2

2

0 = 1

2

−1 =

1

2

2

−2 =

1

4Any multiplication pattern, in reverse, becomes a DIVISION pattern.

When applying the division pattern to exponents, it proves a few things.

1. It shows why a base to the power of 1 is itself.

Example: 5

2 = 25

25 ÷ 5 = 5

Therefore,

= .

You can also think of 5

1

as one 5, but that doesn’t help you extend to the power of

zero or NEGATIVE exponents.

2. It also shows why bases to the power of 0 equal 1.

Example: 5

2 = 25

25 ÷ 5 = 5

5

1 = 5

5 ÷ 5 = 1

Therefore,

= .

*There is an EXCEPTION. If the base is 0, and since you have to use a division

pattern to get to a base with the power of 0, then you have to divide zero by zero.

What do we get when we divide any number by 0? It’s undefined. You cannot

divide by zero. So,

= . Not all mathematicians agree with this,

but for middle school, this is what we can understand and support with division.

3. Any base to the power of a negative exponent is the reciprocal of the same

base to the opposite, positive exponent.

Example: 5

−2 =

1

25

5

2 = 25

Example: 2

−3 =

1

8

2

3 = 8