Find practice problems and solutions for these concepts at Working with Exponents Practice Problems.
This lesson shows you how to add, subtract, multiply, and divide expressions with exponents. You will also learn how to raise expressions to a power.
What Is an Exponent?
An exponent tells you how many times a factor is multiplied. An exponent appears as a raised number that is smaller in size than the other numbers on the page. For example, in the expression 43, the three is the exponent. The expression 43 shows that four is a factor three times. That means four times four times four. Here are examples of exponents and what they mean:
5^{2} = 5 · 5
2^{3}= 2 · 2 · 2
a^{2} = a · a
2x^{3}y^{2} = 2 · x · x · x · y · y
A googol is a 1 followed by 100 zeros. Using exponents, this is written as 10^{100}. The term googol was coined by the eightyearold nephew of American mathematician Edward Kasner in 1938. Kasner asked his nephew what he would name a really large number, and "googol" was the boy's response. 
Adding and Subtracting with Exponents
In previous lessons, when you combined similar terms, you added the numbers in front of the variables (coefficients) and left the variables the same. Here are some examples:
3x + 4x = 7x
2x^{2} + 7x^{2}= 9x^{2}
3xy + 6xy = 9xy
5x^{3} – 3x^{3} = 2x^{3}
What do you do with exponents when you are adding? Nothing! That's right, you add only the coefficients. The variables and their exponents stay the same. This is not a new concept. You used it in previous lessons when you combined similar terms.
Multiplying with Exponents
The rules for multiplying expressions with exponents may appear to be confusing. You treat exponents differently from ordinary numbers. You would think that when you are multiplying, you would multiply the exponents. However, that's not true. When you are multiplying expressions, you add the exponents. Here's an example of how to simplify an expression.
Example:
x^{2} · x^{3}
(x · x)(x · x · x)
x^{5}
You can see that you have 5 x's, which is written as x^{5}. To get x^{5} for an answer, you add the exponents instead of multiplying them.
TipWhen an expression is written as x^{2}x^{5}, it implies multiplication. You do not need to use the multiplication symbol. 
Example:
a^{3} · a^{4}
(a · a · a)(a · a · a · a)
a^{7}
The factors of a^{3} are a · a · a. The factors of a^{4} are a · a · a · a. The factored form of a^{3} · a^{4} is a · a · a · a · a · a · a. When you write the problem out in factor form, you can see that you have 7 a's. The easy way to get 7 is to add the exponents.
What would you do if you see an expression like x^{20}, and you want to multiply it by x^{25}? You can see that writing out the factors of x^{20} · x^{25} would take a long time. Think about how easy it would be to make a mistake if you wrote out all the factors. It is much more efficient and fast to use the rule for multiplying exponents: When you are multiplying, you add the exponents.

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