Welcome back to Professor Baker's Math Class! Today, we are diving deeper into Lesson 6.1 with some targeted review and practice. Exponents are the building blocks for much of what we do in Algebra 2, so mastering these rules now will make future topics much smoother.

Key Concepts: Properties of Exponents

Before tackling the homework, let's refresh our memory on the essential rules you will need to solve the problems on Practice C.

  • Product of Powers: When multiplying like bases, add the exponents.
    $$a^m \cdot a^n = a^{m+n}$$
  • Quotient of Powers: When dividing like bases, subtract the exponents.
    $$\frac{a^m}{a^n} = a^{m-n}$$
  • Power of a Power: When raising a power to another power, multiply the exponents.
    $$(a^m)^n = a^{mn}$$
  • Negative Exponents: A negative exponent indicates the reciprocal. Move the term to the opposite side of the fraction bar to make the exponent positive.
    $$a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n$$

Strategies for Practice C

Your homework focuses on the first half of the worksheet (Questions 1-15). Here is a breakdown of how to approach these two distinct sections:

1. Evaluating Expressions (Problems 1-6)

In this section, you are working with actual numbers. The goal is to find a single numerical answer. Pay close attention to negative exponents on fractions. A great shortcut is to flip the fraction and make the exponent positive:

$$ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} $$

For example, if you see something like problem #4 or #6, flip the inside fraction first, then apply the exponent!

2. Simplifying Expressions (Problems 7-15)

Here, you will deal with variables ($x, y, z$). Remember these three steps for success:

  1. Simplify Coefficients: Treat the regular numbers (coefficients) just like normal fractions or multiplication.
  2. Combine Variables: Use your product and quotient rules to combine $x$'s with $x$'s and $y$'s with $y$'s.
  3. Final Polish: Ensure your final answer contains no negative exponents. If you end up with $x^{-3}$, rewrite it as $\frac{1}{x^3}$.

Pro-Tip for Problem #12: You have a fraction raised to a negative power: $\left(\frac{x^3y^2}{2x^4}\right)^{-2}$. Try simplifying the inside of the parentheses first before applying the outer exponent of $-2$. It usually saves you from dealing with messy numbers later!

Homework Assignment

Please complete the following form the attached worksheet:

  • Worksheet: 6-1 Practice C
  • Problems: #1 through #15

Take your time with the negative signs, as that is the most common place for small errors to occur. Good luck, and see you in class!