Welcome back to class! Today, we are completing our toolkit for handling exponents. In our previous session, we covered the basics of multiplying and dividing with the same base. Now, we are looking at the "special cases" and distribution rules that often trip students up: Negative Exponents, Zero Exponents, and the Power of a Product and Quotient.

Below is a breakdown of these rules based on today's lecture notes.

1. The Zero Exponent Rule

This is one of the most important conceptual hurdles in algebra. It is easy to assume that a number raised to the zero power is zero, but the math tells a different story.

Rule: Any non-zero base raised to the zero power equals 1.

$$x^0 = 1$$

Why? As we saw in the class notes (see Page 3), if you treat a fraction like $\frac{4^5}{4^5}$, basic division tells us the answer is 1. However, using the subtraction rule of exponents, $4^{5-5} = 4^0$. Therefore, $4^0$ must equal 1.

2. Negative Exponents

A negative exponent does not make the number negative. Instead, it indicates a position change (taking the reciprocal). Think of the negative sign as a ticket to switch floors in a fraction.

Rule: If a base has a negative exponent, move it to the opposite side of the fraction line and make the exponent positive.

$$x^{-m} = \frac{1}{x^m} \quad \text{and} \quad \frac{1}{x^{-m}} = x^m$$

For example, looking at our notes, $2^{-3}$ becomes $\frac{1}{2^3}$ or $\frac{1}{8}$.

3. Power of a Product & Quotient

When an exponent sits outside a set of parentheses, you must distribute that power to every factor inside the parentheses.

Power of a Product Rule:

$$(xy)^n = x^n y^n$$

Power of a Quotient Rule:

$$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$

Be careful! As shown in note Page 7, if you have coefficients, they get raised to the power too. For example: $$(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6$$

Homework Assignment

To solidify these concepts, please complete the problems on the attached worksheet.

  • Task: Complete problems #1-73.
  • Focus: Pay close attention to Objective 2 (Zero and Negative exponents) and Objective 4 (Power rules).
  • Tip: Remember to write all final answers with positive exponents only.

Good luck, and see you in the next class!