Welcome back to class! Today, we laid the groundwork for working with exponents. While they might look intimidating at first, exponents are really just a shorthand way of writing repeated multiplication. Once you understand the patterns, simplifying complex expressions becomes much easier.

Before we jump into the rules, let's remember what an exponent actually is. If we look at $2^4$, the $2$ is our base and the $4$ is our exponent. This simply means we multiply 2 by itself 4 times:

$$2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$$

Today we covered the first three essential rules that will help you simplify algebraic expressions.

Rule 1: Product of Powers

The Rule: If the bases are the same and we are multiplying, we add the exponents.

$$x^m \cdot x^n = x^{m+n}$$

Why does this work? Think about $2^3 \cdot 2^4$. That is $(2 \cdot 2 \cdot 2)$ multiplied by $(2 \cdot 2 \cdot 2 \cdot 2)$. If you count them all up, you have seven 2s being multiplied, or $2^7$.

Examples from class:

  • $2^4 \cdot 2^5 = 2^{4+5} = 2^9 = 512$
  • $x^3 \cdot x^8 = x^{11}$
  • $x^5 \cdot y^3 = x^5 y^3$ (Note: We cannot combine these because the bases $x$ and $y$ are different!)

Rule 2: Quotient of Powers

The Rule: If we have the same base and we are dividing, then we subtract the exponents.

$$\frac{x^m}{x^n} = x^{m-n}$$

Why does this work? This is all about "canceling out." If you have $\frac{2^7}{2^3}$, you have seven 2s on top and three 2s on the bottom. Three of them cancel out from the top and bottom, leaving you with four 2s remaining on top.

Examples from class:

  • $\frac{2^7}{2^3} = 2^{7-3} = 2^4 = 16$
  • $\frac{x^5}{x^3} = x^{5-3} = x^2$

Rule 3: Power of a Power

The Rule: If you have a power raised to another power, you multiply the exponents.

$$(x^m)^n = x^{mn}$$

Why does this work? If you have $(x^3)^4$, it means you have the group $(x^3)$ four times: $x^3 \cdot x^3 \cdot x^3 \cdot x^3$. Using Rule 1, we would add $3+3+3+3$, which is the same as $3 \cdot 4$.

Examples from class:

  • $(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64$
  • $(x^4)^3 = x^{12}$

Homework

To master these rules, practice is key! Please complete the attached Exponent Worksheet. It includes sections on finding the value of expressions, simplifying products, and working with quotients. Remember to show your work!