Numbers can be raised to powers more than once. In this section we’ll see what happens when you raise a quantity to a power, and then raise the result to another power.
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120 Powers and Roots
When exponents multiply
Imagine that we have a number a that is not equal to 0. Suppose p and q are integers. What happens if we raise a to the pth power, and then raise the result to the qth power? Mathemati-cally, we get this expression:
(ap)q
It is tempting to suppose that the result of this operation will always produce a huge number.
That can happen if the absolute value of a is larger than 1, and if p and q are both positive and more than 1. For example:
(45)6= 1,0246
= 1,152,921,504,606,846,976
It doesn’t always work out that way, however. If the absolute value of a is between 1 and 0, and if p and q are both positive and more than 1, the number may be quite close to 0. For example,
[(−0.1)3]5= (−0.001)5
= −0.000000000000001
When you have any expression of this sort, you can get the same result if you take the base a to the power of the product of the exponents pq. That is,
(ap)q=apq
Let’s call this the multiplication-of-exponents (MOE) rule. Looking at the numerical examples we just saw, and putting them in this form, illustrates this:
(45)6= 45×6
= 430
= 1,152,921,504,606,846,976 and
[(−0.1)3]5= (−0.1)3×5
= (−0.1)15
= −0.000000000000001
You can evaluate expressions with large exponents quickly by using a calculator with an “x to theyth power” key.
Rational-number powers
When you raise an exponentiated quantity (i.e., anything to a power) to another power, either or both of the exponents can be negative. The MOE rule still applies. In fact, we can let the
exponentsp and q be any rational numbers we want. That gives us the powerful, far-reaching generalized multiplication-of-exponents (GMOE) rule ! If a, p, and q are rational numbers and a≠ 0, then
(ap)q=apq
We now have a way to evaluate an expression where we raise a number to a certain power, and then take a root of the result. Remember that a root is a reciprocal power. So, if we encounter an exponent that takes the form r/s, we can call this the product of r and 1/s, and then use the GMOE rule:
(ar)1/s= ar(1/s)=ar/s
That’s how we’d evaluate the sth root of ar. But it also tells us something more: when we take a base number to a rational-number, noninteger power, it’s the same thing as taking the base to an integer power and then taking an integer root of the result. Remember, a rational num-ber is a quotient of two integers! If we reverse the order of the terms in the above three-way equation, we get
ar/s=ar(1/s)= (ar)1/s
This is a heavy dose of abstract math! Let’s look at a couple of specific cases where integers are raised to rational-number powers. First, this:
106/3= 106×(1/3)
= (106)1/3
= 1,000,0001/3
= 100
If you’re astute, you can solve this a lot quicker by noting that 6/3 = 2, so
106/3= 102
= 100
Usually, rational-number powers aren’t this easy to evaluate. The results often produce num-bers that aren’t even rational. Consider this example:
23/2= 23×(1/2)
= (23)1/2
= 81/2
= 2.8284 ...
This is an endless nonrepeating decimal. It cannot be expressed as a ratio of integers, and is not a rational number. You’ll learn more about these types of numbers in the next chapter.
Multiple Powers 121
122 Powers and Roots
Are you confused?
Let’s review the most important points in this chapter. They can be condensed into six statements.
• If a is any nonzero number and −n is a negative integer, the expression a−n means you should raise a to the power of |n|, and then take the reciprocal of the result.
• If a is any nonzero number and m and n are rational numbers, then am times an is the same as a raised to the power of (m+n).
• If a is any nonzero number and m and n are rational numbers, then am divided by an is the same asa raised to the power of (m−n).
• If a is any nonzero number and p is any nonzero integer, then the pth root of a is the same as rais-inga to the power of 1/p.
• If a, p, and q are rational numbers and a is nonzero, then if you raise a to the pth power and take the result to the qth power, it’s the same as raising a to the power of pq.
• If a, p, and q are rational numbers with a and q nonzero, then if you raise a to the pth power and take theqth root of the result, it’s the same as raising a to the power of p/q.
Here’s a challenge!
What do you get if you take the −5/2 power of 6? Mathematically, evaluate this expression and use a cal-culator to figure out the result to several decimal places:
6(−5/2)
Solution
Let’s apply the GMOE rule to this problem. It can be tricky because of the minus sign, and we have to be sure we remember the difference between negative powers and reciprocal powers. Let’s go:
6(−5/2)= 6−5×(1/2)
= (6−5)1/2
= [1/(65)]1/2
= (1/7,776)1/2
Now it’s time to use a calculator! Remember that the 1/2 power is the same as the square root. First we take the reciprocal of 7,776, getting a decimal point, three ciphers, and a long string of digits. Then we hit the square root key with the string of digits still in the display, getting
0.01134023 ...
The digits go on without end, and there’s no apparent pattern. As things turn out, this is not a rational number.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not represent the only way a problem can be figured out. If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!
1. What happens when a negative number is raised to an even positive integer power? An odd positive integer power?
2. What happens if we
(a) Raise −2 to increasing integer powers, starting with 1?
(b) Do the same thing with a base of −1?
(c) Do the same thing with a base of −1/2?
3. What happens if we
(a) Raise −2 to decreasing negative integer powers, starting with −1?
(b) Do the same thing with a base of −1?
(c) Do the same thing with a base of −1/2?
4. Suppose we come across the following expression. We want to simplify it to a sum of individual terms. How can we do this? Here’s a hint: Use the results of the final
“challenge” in Chap. 5.
(y+ 1)2
5. If we use the same technique as in the previous problem, we can also simplify the following expression. How?
(y− 1)2
6. Use the GAOE rule to prove, in narrative form, that for any number a except 0, and for any rational numbers p, q, and r
apaqar=a(p+q+r)
7. Use the GMOE rule to prove, in the form of an S/R table, that for any number a except 0, and for any rational numbers p, q, and r
[(ap)q]r=apqr
8. Prove, in narrative form, that if x is any number except 0, and if r and s are rational numbers, then
(xr)s= (xs)r
9. Show that if you take the 4th root of any positive number and then square the result, it is the same as taking the square root of the original number.
10. Show that if you take the 6th power of any positive number and then take the cube root of the result, it is the same as squaring the original number.
Practice Exercises 123
124
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