Exponents: Advanced Properties
"Working With Exponents"
This video covers advanced properties of exponents for students who have already learned the basics. It includes negative exponents, zero exponents and multiplying and dividing exponents. The song teaches students how to invert the expression when working with negative exponents. It also explains how to add or subtract the exponents when multiplying or dividing them.
We all know about exponents, right?
We told them a little bit about exponents, but let's get a little deeper. Tell them a little more.
You know an is a, n times,
Like 92 = 9 · 9.
Yeah, you know the exponent basics,
Time to re-up with some new cases.
What if the exponent is negative?
We invert the expression, check the method,
If it’s a-n,
That’s just 1⁄an.
So 4-2 = 1⁄42 = 1⁄16.
Or 3-3 = 1⁄33 = 1⁄27.
Now 40? Hold it son,
a0 = 1.
The answer’s 1, no matter what a is,
40 = 1, don’t debate it.
an is a, n times,
Exponents, where I base my rhymes.
a-n, flip it up!
= 1⁄an, let's begin.
an is a, n times,
Exponents, where I base my rhymes.
a-n, flip it up!
= 1⁄an, till the end.
Yo, that sounds cool and everything... But I wonder, how can we multiply exponents? Hmm. I think I got an idea.
Look, if the base is the same,
We can multiply without changing the game.
Don’t multiply the exponents, add 'em,
Keep the base the same, now that’s the action.
So 23 · 25,
We just add the exponents up real quick.
Yeah, we get 28,
That’s the answer and there's no mistake.
If we want to divide, do the opposite,
Subtract the exponents, that’s all there is to it.
3-2 ÷ 33, now let’s get it.
That’s -2 – 3,
3-5 is what it’ll be.
Which is 1⁄35,
And we do it like that, and we do it like this.
an is a, n times,
Exponents, where I base my rhymes.
a-n, flip it up!
= 1⁄an, let's begin.
an is a, n times,
Exponents, where I base my rhymes.
a-n, flip it up!
= 1⁄an, till the end.
Exponents mean that you multiply the base number by itself, the number of times the exponent tells you to. A shorter way to write this is to use variables. a represents the base number and n represents the exponent, or the number of times you’re multiplying.
Exponents mean that you multiply the base number by itself, the number of times the exponent tells you to. A shorter way to write this is to use variables. a represents the base number and n represents the exponent, or the number of times you’re multiplying.
When you first see a-n, it looks a little crazy--how can you multiply a negative number of times? But if you think about it, it isn’t actually that tough. What’s the opposite of multiplication? That’s right: division! A negative exponent tells you to divide by the base a certain number of times, rather than multiplying. As we learned when we divided by fractions, multiplying a number by its reciprocal is the same thing as dividing. So when you see a-n, just remember that it’s the same thing as 1⁄an.
All you have to do is make the exponent positive, and then you can solve it normally.
3-3 ← It’s negative, so you have to flip it over.
1⁄33 ← Now you have a positive exponent! Find 33, and you’ve got your answer: 1⁄27
a0 always equals 1, so 30 = 1, and 100 = 1, and 10000 = 1, etc.
Why is this the case? The easiest way to see it is like this.
23 · x = 23
We know that for this to be true, x must equal 1. Let's fill in 1 for x and then simplify the equation. We get:
23 · 1 = 23
1 = 23 ÷ 23
1 = 23 − 3
1 = 20
So any number to the zero power equals one.
a0 always equals 1, so 30 = 1, and 100 = 1, and 10000 = 1, etc.
Why is this the case? The easiest way to see it is like this.
23 · x = 23
We know that for this to be true, x must equal 1. Let's fill in 1 for x and then simplify the equation. We get:
23 · 1 = 23
1 = 23 ÷ 23
1 = 23 − 3
1 = 20
So any number to the zero power equals one.
You know that exponents mean multiplication, so when you’re multiplying two exponents, you’re basically just doing more multiplication.
23 · 25
is the same as
(2 · 2 · 2) · (2 · 2 · 2 · 2 · 2)
How many 2’s is that? 8!
Which means 23 · 25 = 28
So if you are multiplying two exponents with the same base, you can just add the exponents to get your answer.
You know that exponents mean multiplication, so when you’re multiplying two exponents, you’re basically just doing more multiplication.
23 · 25
is the same as
(2 · 2 · 2) · (2 · 2 · 2 · 2 · 2)
How many 2’s is that? 8!
Which means 23 · 25 = 28
So if you are multiplying two exponents with the same base, you can just add the exponents to get your answer.
If multiplying two exponents means more multiplication, dividing means doing less multiplication. Instead of adding the exponents up, subtract them.
Another way of writing 3-2 ÷ 33 is 1⁄9 ÷ 27.
1⁄9 ÷ 27 = 1⁄9 · 1⁄27 = 1⁄243
3-2 ÷ 33 = 3-5 = 1⁄35 = 1⁄243
Tests and answer keys are only available for paid subscribers.
Click below to sign up for a digital subscription.
Like us? Like us.
