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Rules of Exponents

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

Summary: This module provides practice problems designed to develop concepts related to the rules of exponents.

Exercise 1

Here are the first six powers of two.

  • 21=2 2 1 2
  • 22=4 2 2 4
  • 23=8 2 3 8
  • 24=16 2 4 16
  • 25=32 2 5 32
  • 26=64 2 6 64
  • a. If I asked you for 27 2 7 (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term?
  • b. IWrite an algebraic generalization to represent this rule.

Exercise 2

Suppose I want to multiply 25 2 5 times 23 2 3 . Well, 25 2 5 means 2×2×2×2×2, and 23 2 3 means 2×2×2. So we can write the whole thing out like this.

An image showing the expanded meaning of an exponent

  • a. This shows that ( 25⁢23=2[ ] 2 5 2 3 2 [ ]
  • b. Using a similar drawing, demonstrate what 103⁢104 10 3 10 4 must be.
  • c. Now, write an algebraic generalization for this rule.
  • d. Show how your answer to 1b (the “getting from one power of two, to the next in line”) is a special case of the more general rule you came up with in 2c (“multiplying two exponents”).

Exercise 3

Now we turn our attention to division. What is 312310312310 size 12{ { {3 rSup { size 8{"12"} } } over {3 rSup { size 8{"10"} } } } } {}?

  • a. Write it out explicitly. (Like earlier I wrote out explicitly what 25⁢23 2 5 2 3 was: expand the exponents into a big long fraction.)
  • b. Now, cancel all the like terms on the top and the bottom. (That is, divide the top and bottom by all the 3s they have in common.)
  • c. What you are left with is the answer. So fill this in: 312310312310 size 12{ { {3 rSup { size 8{"12"} } } over {3 rSup { size 8{"10"} } } } } {} 3[ ] 3 [ ] .
  • d. Write a generalization that represents this rule.
  • e. Suppose we turn it upside-down. Now, we end up with some 3s on the bottom. Write it out explicitly and cancel 3s, as you did before: 310312310312 size 12{ { {3 rSup { size 8{"10"} } } over {3 rSup { size 8{"12"} } } } } {}= ___________________________ = 1313 size 12{ { {1} over {3 rSup { size 8{ left [~ right ]} } } } } {}
  • f. Write a generalization for the rule in part (e). Be sure to mention when that generalization applies, as opposed to the one in part (d)!

Exercise 4

Use all those generalizations to simplify x3y3x7x5y5x3y3x7x5y5 size 12{ { {x rSup { size 8{3} } y rSup { size 8{3} } x rSup { size 8{7} } } over {x rSup { size 8{5} } y rSup { size 8{5} } } } } {}

Exercise 5

Now we’re going to raise exponents, to exponents. What is 234 2 3 4 ? Well, 23 2 3 means 2×2×2. And when you raise anything to the fourth power, you multiply it by itself, four times. So we’ll multiply that by itself four times:

234 2 3 4 = (2×2×2) (2×2×2) (2×2×2) (2×2×2)

  • a. So, just counting 2s, 234=2[ ] 2 3 4 2 [ ] .
  • b. Expand out 1053 10 5 3 in a similar way, and show what power of 10 it equals.
  • c. Find the algebraic generalization that represents this rule.

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