Multiplication Properties of Exponents
An exponent is a mathematical notation that tells how many times a base number should be multiplied by itself. Rules tell us how to add, multiply, subtract and divide with exponents. These laws work when using numbers as well as when using variables, such as X or Y. To multiply two or more base numbers or variables with exponents, there are a few simple rules to follow.
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Quick Review
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A quick review of exponents shows that the exponent tells us to multiply a number by itself that many times. For example, 2 to the third power is the same as 2 x 2 x 2, or 8. Ordinary integers actually have an exponent: one. However, rather than expressing it as 2 to the first power, we simply write 2.
When a variable, such as X or Y, is used in place of a base number, the exponent still indicates how many times that variable is to be multiplied by itself. There's one rule for multiplication with exponents when the bases are the same and another for when the bases are different.
Like Base Numbers or Variables
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When the base numbers are the same in a multiplication problem, add the two exponents together to get the new exponent for the base number. For example, 4 to the fifth power x 4 third power = 4 to the eighth power. Here we have added the two exponents (5 + 3) to get our new exponent (8). The base number remains the same (4).
This also works with variables (X to the second power x X to the third power = X to the fifth power), and when multiplying more than two base numbers (3 to the fourth power x 3 to the sixth power x 3 to the second power = 3 to the 12th power).
If you try a simple equation, you can see this works. Multiply 2 to the third power x 2 to the second power by adding the exponents. Your answer will be 2 to the fifth power or 2 x 2 x 2 x 2 x 2, or 32. If you multiply 2 to the third power x 2 to the second power by simply expanding the notation, (2 x 2 x 2) x (2 x 2), you can see the result will be the same: (8 x 4) or 32. To state this rule with variables: X to the a power x X to the b power = X to the power of (a + b).
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Different Base Numbers
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This does not help us, however, when the bases are not the same. For example, 2 to the third power x 4 to the third power does not allow us to simply add the exponents. In this case, we work out each base number exponent combination individually, and then multiply.
In our example, 2 to the third power = 2 x 2 x 2, so 2 to the third power = 8. Then 4 to the third power = 4 x 4 x 4, or 64. Therefore, 2 to the third power x 4 to third power = 8 x 64. Ultimately then, 2 to the third power x 4 to third power = 512.
Different Base Variables
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When the numbers are variables, there is a way to reduce a problem with exponents with different base numbers if some of the base numbers are the same. The solution is to group like base numbers and then add the exponents.
For example, X to the third power x Y to the second power x Z x X to the second power x Y to the fourth power x Z to the second power can be rewritten by regrouping the like bases: X to the third power x X to the second power x Y to the second power x Y to the fourth power x Z x Z to the second power. Once this is accomplished, you can solve the problem by adding the exponents for each base as follows: X to the fifth power x Y to the sixth power x Z to the third power.
Different Base, Same Exponent
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Finally, let's see what happens when the bases are different but the exponents are the same. For example, X to the third power x Y to the third power. First, let's write out the problem to see how we might later skip a step or two. Our problem can be restated as X x X x X x Y x Y x Y. Using our commutative property, we can rewrite this as XY x XY x XY. This is the same as XY to the third power. Looking back at our initial problem, we can see that X to the third power x Y to the third power = XY to the third power. Restating it as a multiplication law, we have X to the a power x Y to the a power =XY to the a power.
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