Algebra Know-It-ALL

616 Worked-Out Solutions to Exercises: Chapters 1 to 9

3. All of these sets are infinite, and the elements of each set can be completely defined by
   means of an “implied list.” Therefore, the cardinality of every one of these sets is א 0.
   We can pair off any of these sets one-to-one with the set N of naturals. As an optional
   exercise, you might want to show how this can be done. Here’s a hint: Multiply the
   naturals all by 2, 10, 100, or any whole-number power of 10 to create “implied lists”
   for the sets.


4. The original equation is

36 x+ 48 y= 216

When we divide through by 12 on either side, we get

(36x+ 48 y)/12= 216/12

which can be morphed to

(36/12)x+ (48/12)y= 18

and finally to

3 x+ 4 y= 18

That’s as simple as we can get it.

5. To figure this out, note that 18 = 9 × 2. Therefore, according to the power of product
   rule, we have

18 1/2= (9 × 2)1/2

= 9 1/2× 2 1/2

= 3 × 2 1/2

This is a product of a natural number and an irrational number. The nonnegative square

roots of large numbers can often be resolved in this way.

6. The number 83 is prime. If we want to factor this into a product of two natural
   numbers and no remainder, the best we can do is 83 × 1. Therefore, we can’t resolve the
   nonnegative square root of 83 into anything simpler than 831/2. We can also tell that it
   is in the most simplified form because it has no factors that are perfect squares.


7. The ratio of 501/2 to 21/2 is the same thing as the quotient of these two numbers. Note
   that both 50 and 2 are taken to the same real-number power, that is, 1/2. According to
   the power of quotient rule, then, we have

50 1/2/21/2= (50/2)1/2

= 25 1/2

= 5