Same Base, Different Power: Understanding Exponents

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Understanding exponents can sometimes be tricky, especially when dealing with the same base but different powers. Let's break down this concept to make it easier to grasp. — Chia Seeds At Walgreens: Find Them Now!

What Does 'Same Base Different Power' Mean?

When we say 'same base different power,' we're talking about expressions like 2^3 and 2^5. Here, the base is '2' in both cases, but the powers (or exponents) are different (3 and 5, respectively). The base is the number being multiplied, and the power tells you how many times to multiply the base by itself. — Vermont State Police: Latest News & Updates

Simple Examples

  • 2^3 = 2 x 2 x 2 = 8
  • 2^5 = 2 x 2 x 2 x 2 x 2 = 32

Rules for Multiplying the Same Base

The key rule to remember when multiplying expressions with the same base is:

a^m * a^n = a^(m+n)

This means if you're multiplying two exponents with the same base, you simply add the powers together. Let's look at an example: — UPS Store Hays KS: Locations, Services, And Hours

  • 2^2 * 2^3 = 2^(2+3) = 2^5 = 32

Why Does This Work?

Consider 2^2 * 2^3. This is (2 * 2) * (2 * 2 * 2). If you remove the parentheses, you get 2 * 2 * 2 * 2 * 2, which is 2^5.

Rules for Dividing the Same Base

When dividing expressions with the same base, the rule is:

a^m / a^n = a^(m-n)

In this case, you subtract the powers. For example:

  • 2^5 / 2^2 = 2^(5-2) = 2^3 = 8

Why Does This Work?

Consider 2^5 / 2^2 = (2 * 2 * 2 * 2 * 2) / (2 * 2). Two of the 2s in the numerator cancel out with the 2s in the denominator, leaving you with 2 * 2 * 2, which is 2^3.

Power of a Power

Another important rule involves raising a power to another power:

(am)n = a^(m*n)

Here, you multiply the powers. For example:

  • (22)3 = 2^(2*3) = 2^6 = 64

Why Does This Work?

(22)3 means (2^2) * (2^2) * (2^2), which is (2 * 2) * (2 * 2) * (2 * 2). This simplifies to 2 * 2 * 2 * 2 * 2 * 2, which is 2^6.

Examples and Practice

Let's go through a few more examples to solidify your understanding:

  1. 3^2 * 3^4 = 3^(2+4) = 3^6 = 729
  2. 5^4 / 5^2 = 5^(4-2) = 5^2 = 25
  3. (42)2 = 4^(2*2) = 4^4 = 256

Common Mistakes to Avoid

  • Adding Bases: You can only add powers when the bases are the same. 2^2 + 3^2 is NOT equal to 5^4.
  • Incorrectly Applying Rules: Make sure to use the correct rule for multiplication, division, or power of a power.

Conclusion

Understanding how to work with exponents that have the same base but different powers is a fundamental concept in algebra. By remembering the rules for multiplication, division, and power of a power, you can simplify complex expressions and solve equations more efficiently. Keep practicing, and these rules will become second nature!