Welcome back to class! Today, we took a significant leap in our Algebra unit. We have officially moved away from working strictly with numbers (like $2^3$) and added variables (like $x^3$ and $y^5$) into the mix. While seeing letters in math problems can sometimes look intimidating, remember the golden rule: the laws of exponents never change.

Whether the base is a $5$ or an $x$, the logic remains exactly the same. Below is a recap of the key concepts from today's lecture notes and a guide to help you crush the homework.

Key Exponent Rules for Variable Expressions

As we saw in the class notes, simplifying expressions often involves using more than one rule in a single problem. Here is your cheat sheet:

  • The Product Rule: When multiplying terms with the same base, keep the base and add the exponents.
    $$x^a \cdot x^b = x^{a+b}$$
  • The Quotient Rule: When dividing terms with the same base, keep the base and subtract the exponents.
    $$\frac{x^a}{x^b} = x^{a-b}$$
  • Power to a Power: When raising a power to another power, multiply the exponents.
    $$(x^a)^b = x^{a \cdot b}$$
  • Power of a Product: Distribute the exponent to every factor inside the parentheses (including the coefficient!).
    $$(2x^3)^5 = 2^5 \cdot x^{15} = 32x^{15}$$

Handling Negative Exponents

A major focus of today's lesson (and the homework) is dealing with negative exponents. Remember that a negative exponent indicates a reciprocal. You can think of it as a ticket to cross the fraction bar.

If you have $x^{-3}$ in the numerator, move it to the denominator to make it positive. If you have a negative exponent in the denominator, move it up to the numerator.

Example from class: $$\frac{x^{-1}y}{xy^{-2}}$$

To simplify this, we move the $x^{-1}$ down and the $y^{-2}$ up:

$$\frac{y \cdot y^2}{x \cdot x} = \frac{y^3}{x^2}$$

Homework Assignment

Your homework is to complete Algebra 1 Unit 7 Exponent Rules Worksheet #2. There are 40 practice problems plus a bonus question.

Important Requirements:

  1. Show Your Work: As stated on the worksheet, you must show your steps to receive credit. Don't just write the answer!
  2. No Negative Exponents: Your final answer must contain only positive exponents.

Take your time, especially with the problems that have both coefficients and multiple variables (like #21 and #40). Download the notes if you need a refresher on the steps we took in class. Good luck!