Welcome back, class! Yesterday, we laid the groundwork by exploring the individual laws of the mathematical universe regarding powers. Today, we are going to level up. We aren't just looking at one rule at a time anymore; we are looking at putting them together and applying multiple rules to a single problem. This is where "Exponents on top of Exponents" comes into play!
Review: The Properties of Exponents
Before we dive into the complex problems, let's briefly recap the cheat sheet from our class notes. Remember, $a$ and $b$ are real numbers, and $m$ and $n$ are integers.
- Product of Powers: $a^m \cdot a^n = a^{m+n}$ (Add the exponents when bases are the same)
- Power of a Power: $(a^m)^n = a^{m \cdot n}$ (Multiply the exponents)
- Power of a Product: $(ab)^m = a^m b^m$ (Distribute the exponent)
- Negative Exponent: $a^{-m} = \frac{1}{a^m}$ (Move the base to the denominator)
- Zero Exponent: $a^0 = 1$ (Anything to the power of zero is 1, provided $a \neq 0$)
- Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\) (Subtract the exponents)
Combining the Rules
The real challenge—and the fun part—begins when we see expressions like $(x^2 y^3)^5$. According to our notes on the Power of a Product and Power of a Power properties, the exponent on the outside applies to every base inside the parentheses.
$$ (x^2 y^3)^5 = (x^2)^5 \cdot (y^3)^5 = x^{10} y^{15} $$
Notice how we multiplied the outer exponent (5) by the inner exponents (2 and 3). Be careful with negatives! As shown in your notes, if you have negative exponents, remember to move them to the denominator to make them positive.
Classwork: 6-1 Integer Exponents Packet
For today's Classwork and Homework, you are to complete the 6-1 Integer Exponents Packet. This packet guides you through:
- Exploring Patterns: Look at the table on the first page. Notice how $5^3 = 125$, $5^2 = 25$, and $5^1 = 5$. What happens when you divide by 5 again? You get $5^0 = 1$. Keep following that pattern to understand negative exponents!
- Practice Problems: You will apply the properties to simplify expressions like $15^2 \cdot 15^{-5}$ and find missing exponents.
- Real-World Application: You'll see word problems involving volume and mass (like the mass of an Emperor Scorpion!).
Discussion Question of the Day
The other day, I asked you to pick a formula for your shop class or a real-world hobby that utilizes exponents. Today, I want you to take that a step further:
Give me a specific example of where that formula is used and explain how the exponent rules might be helpful in solving it.
Example: If you chose the area of a circle ($A = \pi r^2$), perhaps you are calculating the cross-section of a pipe. If the radius is doubled ($2r$), how does that affect the area? Using our exponent rules, $A = \pi (2r)^2 = \pi 4r^2$. The area quadruples!
Check the attached PDF notes for step-by-step solutions to the examples we did in class, including order of operations with exponents. Good luck, and let's get solving!