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Multiplying Exponents With the Same Base
Multiplying exponents with the same base is easier to do than it seems, especially assuming that you understand the basic idea of exponents. For example, take the following problem: 3^2 X 3^3. If you understand that 3^2 is the same thing as 3 X 3, and that 3^3 is the same thing as 3 X 3 X 3, then it makes sense that the original problem would be the same thing as (3 X 3) X (3 X 3 X 3). If you remove the parentheses, you’ll see that you could write that as 3 X 3 X 3 X 3 X 3 – or just 3^5. So 3^2 X 3^3 = 3^5.
Instead of writing all of that out each time you want to multiply exponents, just think of it like this: two threes times three threes is just five threes. You could get the same answer just by adding the powers. So that’s the rule for multiplying exponents – To multiply exponents with the same base, just add the powers.
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Dividing Exponents With the Same Base
Dividing exponents with the same is just as easy as multiplying them. For example, take the following problem: 3^3 / 3^2. The numerator (top of the fraction) is 3^3, which is really the same as 3 X 3 X 3. The denominator (bottom of the fraction) is 3^2, which is really the same as 3 X 3. Now think about fractions. If you have 3 X 3 X 3 on the top of the fraction, and 3 X 3 on the bottom of the fraction, you can simplify the fraction by crossing out two threes in the numerator and two threes in the denominator. The result would be 3^1, or just 3.
How could you have gotten that answer more easily? Well, you know that if there are more threes in the numerator, you can simplify the fraction by crossing out the number of threes that are in the denominator. How can you figure out how many threes that would leave in the numerator? Subtract the number of threes in the denominator from the number of threes in the numerator. So that’s the rule for dividing exponents – To divide exponents with the same base, just subtract the powers.