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Topic 9 Exponentials - Olympic College Topic 9 Exponentials...

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Olympic College Topic 9 Exponentials Page | 1 Topic 9 Exponentials Definition: In Mathematics an efficient way of representing repeated multiplication is by using the notation a n , such terms are called exponentials where a is called the base and n the exponent . In general we say that a n is a to the power of n but in particular when the exponent is 2 we often use the word squared and when the exponent is 3 we use the word cubed. For example, n terms Here are some examples of exponentials 5 3 = = 125 (called 5 cubed) 10 2 = = 100 (called 10 squared) 2 5 = = 32 (called 2 to the power of 5) 1. Properties of Exponentials with the same base. There are three basic operations involving exponentials they are as follows. Multiplication Rule = Division Rule = Powers Rule = We can give a simple explanation of how these rules work by giving specific examples of each situation. These are not formal proofs but they do give a little insight to why the above rules are true. = = = = = = = = = = = =
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Olympic College Topic 9 Exponentials Page | 2 From the above definitions we can conclude the following properties are also true. a 0 = 1 any positive number to the power of zero is 1. any positive number to the power of 1 is the same as its reciprocal. any positive number to the power of n is the same 1 divided by a n . 1 divided by a to a negative power is the same as a to a positive power. Again we can give a simple explanation of how these properties are true. They are not formal proofs but they do give a little insight to how they work. For example, = 7 4-4 = 7 0 = = 1 So 7 0 = 1 = 7 4-5 = = = So 7 1 = = = = 7 5 So = 7 5 = = = 7 4 So = 7 4 Example 1: Simplify the following expressions. (The answers are to have only positive exponents). (a) (b) (c) (d) (e) (f) Solution (a): = Using the multiplication rule. = Solution (b): = Using the multiplication rule. = = = Using the property that
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Olympic College Topic 9 Exponentials Page | 3 Solution (c): = Using the multiplication rule. = Solution (d): = Using the multiplication rule. = = = Using the property that Solution (e): = Using the multiplication rule. = = = 1 Using the property that Solution (f): = Using the multiplication rule. = = = Using the property that Example 2: Simplify the following expressions.
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