Welcome back to class! We have spent some time mastering the individual laws of exponents, but today we are taking it to the next level. What happens when you face a problem that requires three, four, or even five different rules to solve?

It can look intimidating, but if you follow a structured order of operations, these problems become much more manageable. We aren't just memorizing rules today; we are learning the strategy of how to combine them.

The Toolbox: A Quick Review of the Rules

Before we dive into the strategy, let's make sure we have our tools ready. Here are the rules we discussed in class (referenced in the attached notes):

  • Rule 1 (Product Rule): When multiplying same bases, add exponents. $x^m \cdot x^n = x^{m+n}$
  • Rule 2 (Quotient Rule): When dividing same bases, subtract exponents. $\frac{x^m}{x^n} = x^{m-n}$
  • Rule 3 (Power Rule): Power raised to a power? Multiply them. $(x^m)^n = x^{mn}$
  • Rule 4 (Zero Exponent): Anything to the zero power is 1. $x^0 = 1$
  • Rule 5 (Negative Exponents): Negatives switch places (top to bottom or bottom to top). $x^{-m} = \frac{1}{x^m}$
  • Rules 6 & 7 (Distributive Property): If an exponent is outside a parenthesis, distribute it to all parts inside. $(xy)^m = x^m y^m$

The Strategy: How to Use Multiple Rules

When you look at a complex problem, it helps to have a checklist. Based on our class discussion, here is the 4-Step Method to solving these without errors:

  1. Get rid of Parentheses: Start by using Rules 6, 7, and 3. If you see an exponent outside a group, distribute it and multiply powers first. Clear the brackets!
  2. Deal with Multiplication: Look Left and Right in the problem. If you see the same base being multiplied, combine them using Rule 1 (add the exponents).
  3. Deal with Division: Look Up and Down in the problem. If you have the same base on the top and bottom, combine them using Rule 2 (subtract the exponents).
  4. The Cleanup: Finally, look for any Zero or Negative exponents. Change $x^0$ to 1, and flip any negative exponents to the opposite side of the fraction bar (Rules 4 and 5).

Example from Class

Let's look at a problem from the notes to see this in action:

$$ \left(\frac{3}{5}\right)^{-2} $$

Using our strategy, we distribute the exponent to both parts (Rule 7):

$$ \frac{3^{-2}}{5^{-2}} $$

Then, we handle the negative exponents by flipping their positions (Rule 5):

$$ \frac{5^2}{3^2} $$

And finally, we evaluate the numbers:

$$ \frac{25}{9} $$

Class Materials

Below you will find the PDFs containing the handwritten notes on all 7 rules, the worked-out examples from the whiteboard, and the scanned homework page.

  • Class Notes: Detailed breakdown of Rules 1-7.
  • Class Work: Step-by-step solutions for the in-class practice problems.

Homework Assignment

Please complete the following problems from the textbook (Page 326). Focus on Evaluating Numerical Expressions.

Assignment: Page 326 #16-31 (all)

Tip: Remember that when you have a negative base raised to an even power, the result is positive. If it is raised to an odd power, the result remains negative!