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A Bunch of Other Stuff About Radicals

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample word problems which develop concepts related to further uses of radicals.

Let’s start off with a bit of real life again, shall we?

Exercise 1

Albert Einstein’s “Special Theory of Relativity” tells us that matter and energy are different forms of the same thing. (Previously, they were thought of as two completely different things.) If you have some matter, you can convert it to energy; if you have some energy, you can convert it to matter. This is expressed mathematically in the famous equation E = m c 2 E=m c 2 , where E E is the amount of energy, m m is the amount of matter, and c c is the speed of light. So, suppose I did an experiment where I converted m m kilograms of matter, and wound up with E E Joules of energy. Give me the equation I could use that would help me figure out, from these two numbers, what the speed of light is.

Exercise 2

The following figure is an Aerobie, or a washer, or whatever you want to call it—it’s the shaded area, a ring with inner thickness r 1 r 1 and outer thickness r 2 r 2 .

Figure 1
Aerobie
A picture of a washer, or Aerobie
  • a. What is the area of this shaded region, in terms of r 1 r 1 and r 2 r 2 ?
  • b. Suppose I told you that the area of the shaded region is 32 π 32π, and that the inner radius r 1 r 1 is 7. What is the outer radius r 2 r 2 ?
  • c. Suppose I told you that the area of the shaded region is A A, and that the outer radius is r 2 r 2 . Find a formula for the inner radius.

OK, that’s enough about real life. Let’s try simplifying a few expressions, using the rules we developed yesterday.

Exercise 3

100 y 100 y

Exercise 4

x 25 x 25

Exercise 5

x + 16 x + 16

Exercise 6

50 2 = 50 2 =

Exercise 7

5 2 + 2 3 - 3 2 = 5 2 +2 3 -3 2 =

Exercise 8

27 - 48 = 27 - 48 =

Let’s try some that are a bit trickier—sort of like rational expressions. Don’t forget to start by getting a common denominator!

Exercise 9

1 2 + 2 2 = 1 2 + 2 2 =

  • a. Simplify. (Don’t use your calculator, it won’t help.)
  • b. Now, check your answer by plugging the original formula into your calculator. What do you get? Did it work?

Exercise 10

1 2 + 1 3 = 1 2 + 1 3 =

  • a. Simplify. (Don’t use your calculator, it won’t help.)
  • b. Now, check your answer by plugging the original formula into your calculator. What do you get? Did it work?

Exercise 11

1 3 + 1 - 3 2 = 1 3 + 1 - 3 2 =

  • a. Simplify. (Don’t use your calculator, it won’t help.)
  • b. Now, check your answer by plugging the original formula into your calculator. What do you get? Did it work?

And now, the question you knew I would ask…

Exercise 12

Graph y = x y= x

  • a. Plot a whole mess of points. (Choose x-values that will give you pretty easy-to-graph y-values!)
  • b. What is the domain? What is the range?
  • c. Draw the graph.

Exercise 13

Graph y = x - 3 y= x -3 by shifting the previous graph.

  • a. Plug in a couple of points to make sure your “shift” was correct. Fix it if it wasn’t.
  • b. What is the domain? What is the range?

Exercise 14

Graph y = x - 3 y= x - 3 by shifting the previous graph.

  • a. Plug in a couple of points to make sure your “shift” was correct. Fix it if it wasn’t.
  • b. What is the domain? What is the range?

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