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Exponents and Factors

Do you remember the section on factors? Do remember how we broke numbers apart with prime factorization? Here's an example.

Factors of 16: 1*16, 2*8, 4*4
Prime factors of 16: 2, 2, 2, 2
Prime factorization of 16: 2 * 2 * 2 * 2 = 16

We're going to jump into the idea of exponents and roots in this section. Exponents are easy. They are just a quick way of writing down long multiplication problems with the same values.

2 * 2 * 2 * 2 = 16 We're multiplying the number 2 four times in a row. Mathematicians sometimes have to multiply ten or twenty 2's in a row. They don't want to write it out so they write...

2⁴ = 16

2 is the base and 4 is the exponent. The 4 represents the number of times you multiply 2 by itself.

3⁶ = 3 * 3 * 3 * 3 * 3 * 3 = 729
Prime factors of 729: 3, 3, 3, 3, 3, 3

What about 24?
Prime factors of 16: 2, 2, 2, 3
We're going to have to break it up a bit since we don't have all 2's.
2³ * 3 = 24

You'll read about squares and cubes when doing math. Squared is a number that has the exponent=2. A cubed number is one that has an exponent=3.

'0' and '1' are special exponents. When the exponent is '1' the number is itself. It's just writing the number down once and multiplying it by no other terms. When the exponent is '0' The value is '1'. You'll get more detailed explanations later. Just remember that any number with an exponent equal to '0' is one.

6¹ = 6
2,578¹ = 2,578

6⁰ = 1
2,578⁰ = 1

Looking at the Roots

Since we're talking about exponents we are also introducing roots. You'll hear about square roots all the time. Cube roots too. There are as many roots as there are exponents. A root is a bit like a factor because a number can be broken down into its roots. They work in a reciprocal way to exponent values. Here are some different ways the problem might be given.

What is the square root of 4?
What is the second root of 4?

Answer: 2

The answer is two because a square root asks what two equal factors are multiplied to make the number.

What are the two equal factors of 4?
Answer: Factors of 4: 1*4, 2*2... so you write down 2 because it is one of two equal factors.

What is the square root of 9?
Factors of 9: 1*9, 3*3
The square root of 9 is 3 because it is one of the two equal factors.

Examples of square roots (and factors):
2 is the sqrt of 4 (1*4, 2*2)
3 is the sqrt of 9 (1*9, 3*3)
4 is the sqrt of 16 (1*16, 2*8, 4*4)
5 is the sqrt of 25 (1*25, 5*5)
6 is the sqrt of 36 (1*36, 2*18, 3*12, 6*6)

We'll slow down for now, but take a look at 4. So we know that 2 is the square root of 4 and 4 is the square root of 16. How are 2 and 16 connected?

2 * 2 = 4
4 * 4 = 2 * 2 * 2 * 2 = 16
From before... 2⁴ = 16
That means that the fourth root of 16 is 2 because four twos can be multiplied out to be 16. Two is one of four equal factors that make up 16.

It's harder to do bigger numbers in your head, but you will need to answer questions about roots and exponents as you move forward in math. Don't worry. The way you solve the problem always breaks down to factors. We do calculus all the time and we use these simple factoring rules every day.

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