Understanding the Rules of Exponents
Welcome, students! Today, we’re going to dive into the rules of exponents. These rules are essential for simplifying expressions and solving equations that involve exponential terms.
What are Exponents?
An exponent refers to the number that indicates how many times a base is multiplied by itself. For example, in 34, the number 3 is the base and the number 4 is the exponent, which means 3 is multiplied by itself 4 times: 3 * 3 * 3 * 3.
Basic Rules of Exponents
1. Product Rule
The product rule tells us that when we multiply two powers with the same base, we can simply add the exponents. For example:
am * an = am+n
Example: 23 * 24 = 23+4 = 27
2. Quotient Rule
The quotient rule states that when we divide two powers with the same base, we subtract the exponents. For example:
am / an = am-n
Example: 56 / 52 = 56-2 = 54
3. Power Rule
The power rule is used when raising a power to another power. In this case, you multiply the exponents. For example:
(am)n = amn
Example: (32)3 = 32*3 = 36
4. Zero Exponent Rule
Any base with an exponent of zero is equal to one. For example:
a0 = 1 (provided that a ≠ 0)
Example: 70 = 1
5. Negative Exponent Rule
A negative exponent indicates that the base is on the wrong side of a fraction and should be flipped to the other side. For example:
a-n = 1 / an
Example: 2-3 = 1 / 23 = 1/8
Questions You Might Have
- What happens if the bases are different?If the bases are different, you cannot simplify the expression using the product or quotient rules. Those rules only apply to like bases.
- Can these rules be applied to variables as well as numbers?Yes, these rules apply to any expression with an exponent, whether it contains variables, numbers, or both.
Problems to Solve
Let’s try applying these rules with some practice problems.
- Simplify: x5 * x3
- Simplify: y7 / y2
- Simplify: (z4)2
- Write with positive exponents: a-3
- Simplify: 100
Solutions
Here’s how you would solve each problem:
- x5 * x3 = x5+3 = x8
- y7 / y2 = y7-2 = y5
- (z4)2 = z4*2 = z8
- a-3 = 1 / a3
- 100 = 1
Remember, practice makes perfect when it comes to mastering these rules. Keep working through problems, and you’ll get the hang of it!
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