166 Review Questions and Answers
Question 8-10
Suppose we have a nonzero number x, an integer p, and a nonzero integer q. We want to take
theqth root of the quantity xp. How can we express this as a single power of x?Answer 8-10
Let’s remember that the qth root of any quantity is the same as the (1/q)th power. This lets us
apply the rule from Answer 8-9, like this:(xp)1/q=xp(1/q)
=xp/qWe divide p by q and then raise x to that power. It’s important that q never be 0! If it is, we’ll
get an exponent of p/0, and the entire expression will become meaningless.Chapter 9Question 9-1
No matter how close together two rational numbers happen to be, we can always find another
rational number between them. How?Answer 9-1
We can take the average of the original two numbers. If those two numbers are p and q, then
the rational number ( p+q)/2 is always greater than p but less than q.Question 9-2
Imagine that we take a continuous, infinitely long geometric line and make it into a number
line. Every rational number will then correspond to a unique point on this line. Does that
mean every point on the line will represent a rational number?Answer 9-2
No! Even though every rational number can be represented on the line, there are “extra” points
on the line that don’t correspond to any rational number.Question 9-3
How many points on a true geometric number line do not correspond to any rational number?
Give two examples.Answer 9-3
There are infinitely many such points. A good example is the point corresponding to the
positive square root of 2. We saw how that point can be located in the text of Chap. 9.
Another example is π, the ratio of a circle’s circumference to its diameter. We can locate
this point by taking a circle with a diameter of exactly 1 unit, and then “rolling” it for one
complete revolution, without slipping or sliding, upward along the number line as shown
in Fig. 10-4.