4.1 Rules of Exponents

Lesson Plan 4.1: Rules of Exponents

Objectives:

• Understand meaning of exponents.
• Become familiar with exponent rules, especially in the context of numerical bases.
• Become comfortable applying exponent rules to all bases.

Suggested Lecture Breaks:

• MWF: You have three days.
• MW/TR: You have two days.

You're allotted 2-3 days on this section. Break lecture into 2-3 roughly equal chunks. At the start of the second (and third) class day, it might be a good idea to recap all the rules they've learned previously. You might do this by asking students to provide rules or examples given the name of the rule.

Note that the first two days (if you're teaching a MWF class) and the first day (if you're teaching a MW/TR class) need to go over all the basic exponent rules: product, power, negative exponent definition, and the quotient rule, as these are all covered in that week's WebWork assignments.

Suggested Lecture Notes:

• Introduce the notion of exponents as a shorthand for repeated multiplication. (It may be useful to note that we use multiplication as a shorthand for repeated addition.)
• Introduce each exponent rule with a motivating example, such as the one below, as a way to show why the product rule makes sense
  • $2^5 \cdot 2^3 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^8 = 2^{5+3}$
• For each rule give an additional example that is impractical to write out, such as:
  • $7^{15} \cdot 7^{110} = 7^{15+110} = 7^{125}$
• Provide a few examples in which students must simplify using the exponent rules, include variables at this stage.
• For each rule give an example such as
  • $(\frac{-2y^3z^{-1}}{z^4})^6$
• Explain why we can't 'distribute' exponents over additional parentheses:
  • $(x+y)^2 \neq x^2 + y^2$
• Emphasize what it means for an expression to be simplified as much as possible.

Comments on the handout:

• Remind students about order of operations and to look carefully at what exponent is applied to.
• Since we are spending multiple days on this section, encourage students to focus on the problems with numerical bases so that they become more comfortable with the rules without being intimidated by notation yet.
• Question 7: Be sure to emphasize no calculators as the problem says.
• Question 8d: Students often try to ``distribute" the exponent. This may be a good one to go over as a class.
• Students can never get too much practice with exponent problems. If students complete the worksheet for this section, provide a handout with additional problems.