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## What are Exponents?

Let’s do a group of multiplying.

3 × 3 = 9

3 × 3 ×3= 27

3 × 3 × 3 × 3 = 81

3 × 3 × 3 × 3 × 3 = 243

3 × 3 × 3 × 3 × 3 × 3 = 729

What if we wanted to multiply ten 3’s together? I sure do not want to have to write that all out.

What if it was a hundred 3’s? So, we’ll take a trick to find rapidly.

3 × 3 × 3 × 3 = 3^{4} = 81

3 × 3 = 3^{2} = 9

3 × 3 × 3 = 3^{3} = 27

3 × 3 × 3 × 3 = 3^{4} = 81

3 × 3 × 3 × 3 × 3 = 3^{5} = 243

We can do this with any number…

7 × 7 × 7 × 7 × 7 = 16,806

5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 = 5^{10} = 9,765,625

Your Turn:

Expand out 2^{10} and calculate answer.

Here are some mathematical terms to learn:

image

Now check ordinary number. What happens when the exponent is 1?

2^{1}= 2

Only 2

9^{1}= 9

2198252^{1}= 2198252

What about if exponent 0?

We generate a special rule for it

2^{0}= 1

3^{0}= 1

4^{0}= 1

Any number to the zero power is 1.

Your Time:

5^{0}

312^{0}

89469^{0}

## Important Exponents to Know

When you’re doing math – especially during algebra, you’ll be able to know some important exponent from the top of your head.

### Perfect Squares

These are from your multiplication table … So, you should already know this. If you don’t, spread those flashcards and go for it!

1^{2} = 1, 2^{2} = 4, 3^{2} = 9

4^{2} = 16, 5^{2} = 25, 6^{2} = 36

7^{2} = 49, 8^{2} = 64, 9^{2} = 81

10^{2} = 100, 11^{2} = 121, 12^{2} = 144

13^{2} = 169, 14^{2} = 256, 15^{2} = 225

16^{2} = 256, 17^{2} = 289, 18^{2} = 324

Why are they called “perfect squares” (or simply “squares”)? Because these are the areas of a square!

### Perfect Cube

1^{3} = 1, 2^{3} = 8, 3^{3} = 27

4^{3} = 64, 5^{3} = 125, 6^{3} = 216

7^{3} = 343, 8^{3} = 512, 9^{3} = 729

I’ll bet you can guess why these are called “cubes”? Yes, they are cubes!

You also need to know a chunk of the powers of 2…

2^{2} = 4, 2^{3} = 8, 2^{4} = 16

2^{5} = 32, 2^{6} = 64, 2^{8} = 256

Some of 4’s…

4^{1} = 4, 4^{2} = 16, 4^{3} = 64

Some of 10’s…

10^{1} = 10, 10^{2} = 100, 10^{5} = 10000

## Exponents with Negative Bases

Recall some terms:

Image

The first place where negative numbers can be displayed is in the base.

What I’m going to show you now is that this is a really common place to mess with algebra … So, pay extra attention!

Are these the same?

( -2) ^{2} and -2^{2}

These are actually very different … So, let’s learn what each of us means!

( -2) ^{2}

The parentheses sign is important! They put a box around the negative!

( -2) ^{2} = (-2) (-2) = 4

The sign minus*minus plus i.e. – * – = +

Remember that a negative time is a negative of a positive!

Other one:

-2^{2}

This negative is out in front and not engaged into the 2.

-2^{2} = -2 × 2 = -4

The negative sign stays out in front.

What if we did cubes with negative bases?

( -2) ^{3} and -2^{3}

Are these going to be the same?

Let’s just write down what each one means and see what happens.

( -2) ^{3} = (-2) (-2) (-2) = -8

The parenthesis sign means that the negative is engaged in the box with the 2.

-2^{3} = -2 × 2 × 2 = -8

The negative sign stays out in front.

Hey, both answers are negative this time! So, this is the same!

The key is to always remember what it means and what is going on and you can never go wrong.

## Negative Exponents

The other place negative numbers can pop up in these critters is in the exponent or power, itself!

What kind of math is that? Okay, we know

2^{2} = 25

So, what does it mean when the exponent or power is negative? It means that we have to do something exceptional…

2^{-2} = ½^{2}

We make a fraction with the whole thing in the denominator…

and make the exponent positive!

Yes really. That’s what you do. Sounds weird though!

Let’s finish it:

2^{-2} = ½^{2} = ¼

Here’s another one:

2^{-3}

Just do the same thing! Put the whole one at the bottom of a fraction with the top one … and finish it.

2^{-3} = ½^{3} = ⅛

So, what happens if it’s already a fraction with a negative exponent?

½^{-2}

Just do the reverse!

½^{-2} = 2^{2}

I know this lesson probably doesn’t make sense properly. In general, we can remember the things in mathematics when we understand what is going on – and this is the best way. But for that you have to memorize how it works.