DEFINITIONS
Exponents look like this:
23
The big number, 2, is called the base.
The little number, 3, is called the exponent.
The exponent tells us how many times we’re going to multiply the base by itself.
So 23 = 2*2*2
When we have exponents with variables, like this:
4x3
Coefficient: the number in front (4)
Variable/base: the part in the middle (x)
Exponent: little number in the corner (3)
If you have a term like 4x, with no exponent visible, the exponent is 1.
Like Terms must have the same variable with the same exponent.
EX: 3x3 , -2x3 and 5x3 are like terms because they have the same variable (x) and the same exponent (3).
EX: 3x2 3x and 3x3 are NOT like terms because they have different exponents.
ADDING AND SUBTRACTING
You can only add and subtract like terms. Otherwise, you’re adding apples and oranges.
When adding, we do not change the exponents.
Ex: Simplify
7x2 – 3x2
4x2
This is saying we have 7 x-squareds and we’re taking away 3 x-squareds, so we have 4 x-squareds left.
Ex: Simplify
4x3 + x2 – 5 + x3 – 2x2
We combine like terms:
4x3 + x3 = 5x3
x^2 + x^2 = 2x^2
5 has no like term, so we leave it alone.
So we get: 5x3 + 2x2 – 5
(Usually, we write polynomials so that the term with the biggest exponent comes first. Then the exponents get smaller and smaller as you go to the left.)
MULTIPLYING AND DIVIDING
You don’t need like terms to multiply or divide. Here, the exponents can change.
MULTIPLYING
To understand how to multiply with exponents, we have to remember what an exponent does.
Ex: Simplify
x2 * x3
Remember, x2 means x*x and x3 means x*x*x, so we can rewrite the problem as
x*x * x*x*x
Which we can write as x5
Notice, we are adding the exponents
Ex: Simplify
3x2y^3(4x5y^9)
First, multiply the constants: 3*4 = 12
Then we see how many x’s we are multiplying together. There are 2 in the first part and 5 in the second part, so altogether we are multiplying 7 x’s together, which is x7
There are 3 y’s in the first part and 9 y’s in the second part, so altogether, we have 12 y’s being multiplied, or y12
So our answer is: 12 x7y12
Ex: Simplify
6x3y4z(2x4yz2)
=12x^7y^5z^3
MULTIPLICATION RULE: Add the exponents of the same variable
DIVIDING
Dividing with exponents involves a lot of “canceling.” Again, we have to remember what the exponents mean.
EX: Simplify
y5
y3
This means
y*y*y*y*y
y*y*y
We can cancel three of the y’s in the top with the three y’s in the bottom, so we’re left with
y*y or y2
Notice, we take away the bottom exponent from the top exponent. This is subtracting.
DIVISION RULE: Subtract the bottom exponent from the top exponent of the same variable.
OTHER EXPONENTS
The division rule leads us to two important things in exponents.
First
x2
x2
There are two ways to think about this. One is to cancel x2 with x2 to get 1.
The other is to use the division rule of subtracting exponents to get x0.
Both answers are correct, so we see that
x0 = 1
Second
x2
x5
Again, there are two ways to look at this. If we use canceling, we get
1
x3
If we use the division rule, we get x-3
Again, both are valid representations of the original expression.
So we see that
1
x3 = x-3
This is true for any exponent.
In fact, a negative exponent will always change the position of the variable it affects.
EX: Simplify
4
x-6
The negative in the exponent moves the variable to the top of the fraction
4x6
POWER AND PARENTHESES
An exponent on the outside of parentheses is going to do something interesting.
EX: Simplify
(x5y2)3
OK. The exponent on the outside of the parentheses is telling us to multiply what’s inside the parentheses by itself three times, like this:
x5y2 * x5y2 * x5y2
Now, using our multiplication rule, we get x15y6
See how we got x5 three times and y2 three times? We are multiplying.
PARENTHESES RULE: An exponent outside the parentheses will multiply with the powers inside the parentheses.
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