Algebra Know-It-ALL

140 Irrational and Real Numbers

is irrational. But you can “zoom in” on it. Divide out the fractions 59/19 and 60/19 with a calculator to

obtain their decimal expansions. These are both rational numbers, and you’ve learned how you can take

any number to a rational-number power. If you expand them to five decimal places, and then display π

to the same number of places (using a calculator with a π key), you will see that π is between these two

rationals:

59/19 = 3.10526 ...

π= 3.14159 ...

60/19 = 3.15789 ...

That means

59/19<π< 60/19

You should now be able to imagine, without too much trouble, that

2 59/19< 2 π< 2 60/19

The “mystery number” is in that interval somewhere. It’s a real number, and it corresponds to its own

special point on a real-number line. You can indeed raise a number, variable, or expression to an irrational

power.

Here’s a challenge!

Suppose x, y, andz are real numbers, with x≠ 0 and y≠ 0. Show that if you take x to the yth power and

then take the result to the zth power, you do not necessarily get the same result as if you take x to the

power of yz.

Solution

All we have to do is find a numerical example where the two expressions don’t agree. There are plenty! Let

x= 2, y= 9, and z= 1/2. Then xy= 29 = 512. If we raise 512 to the 1/2 power, we get 22.627 ..., an endless,

nonrepeating decimal. Now consider yz. That’s 91/2, or 3. If we raise x, which is 2, to the power of 3, we get 8.

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. Which of the following quantities can we reasonably suspect is irrational?
   (a) 163/4
   (b) (1/4)1/2
   (c) (−27)−1/3
   (d) 271/2