1. Introduction
If a is any number and n any positive integer (whole number) then the product of a with itself n times, a1 × a2 × ... × an, is called a raised to the power n, and written an, i.e.:
Example 1:
The following important rules apply to powers:
| Rule 1 | = | ||
|---|---|---|---|
| Rule 2 | = | ||
| Rule 3 | = | ||
| Rule 4 | = | ||
| Rule 5 | = |
We want these rules to be true for all positive values of a and all values of m and n. We shall first look at the simpler cases:
Example 2:
Exercise 1: Simplify each of the following.
Solution: 23 × 23 = 2(3+3) = 26 = 64
Solution: 315 ÷ 312 = 3(15−12) = 33 = 27
Solution: (102)3 = 10(2×3) = 106 = 1,000,000
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2. Negative Powers
The question now arises as to what we mean by a negative power. To interpret this, note that
If Rule 2 is to apply, then a2÷a5 = a2 − 5 = a−3. Thus a−3 = 1 ⁄ a3
The general rule is:
| a−n = 1 ⁄ an |
Example 3:
Exercise 2: Write each of the following in the form ak, for some number k.
Solution: 23 × 2−5 = 23−5 = 2−2, which is 1⁄4.
Solution: 35 ÷ 37 = 35−7 = 3−2, which is 1⁄9.
Solution: (102)−3 = 10(2 × (−3)) = 10−6, which is 1⁄1,000,000.
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3. Fractional Powers
If a is positive, then the square root of a is a number which, when muliplied by itself, gives a. Thus 3 is the square root of 9, since 32 = 9. We write 3 = √9. Note that, by definition, √a × √a = a. This gives us a way of interpreting a1⁄2 for, by Rule 1:
- so that a1⁄2 = √a. The general rule is that, if a is a positive number and n is a positive integer, then:
| a1⁄n = n √a |
- where n √a is the n th root of a. We can see this in general for, by Rule 3:
Example 4:
Note that in Example 4(c) we used Rule 3, i.e. (a1⁄n)m = a1⁄n × m = (a1⁄m)n , so:
Quiz 1: To which of the following does (85)1⁄3 simplify?
Explanation: Using Rule 3, we have:
4. Use of the Rules of Simplification
In this section we shall demonstrate the use of the rules of powers to simplify more complicated expressions.
Example 5:
Beginning with the innermost bracket, we have, using Rule 3:
Then:
Beginning again with the innermost bracket, and using Rule 3, we have:
Now if we use Rule 3 again we have:
We have:
- using Rule 3. Now we use Rule 1:
Starting with the first term:
Thus:
Similarly:
so that:
Finally we have:
The first term simplifies as follows.
Treating the second term:
Thus:
5. Quiz on Powers
Simplify the expressions and choose the solutions from the options given.