Math Example--Exponential Concepts--Laws of Exponents: Example 19

Topic

Exponents

Description

This example demonstrates the simplification of an expression involving multiplication and division with both positive and negative exponents. The problem involves simplifying (x⁶ * x^-10) / x², which results in x^-6 or 1/x⁶. This showcases how positive and negative exponents interact when multiplying and dividing terms with the same base, and how the result can be expressed as a fraction.

Understanding how to handle expressions with mixed positive and negative exponents in both multiplication and division is crucial in algebra and higher mathematics. These examples help students grasp the concept that when multiplying terms with the same base, exponents are added, and when dividing, they are subtracted, regardless of their signs. By presenting various scenarios, students learn to apply these rules confidently in different mathematical contexts.

Exposure to multiple worked-out examples is essential for students to fully comprehend the concept of combining multiplication and division with mixed exponents. Each example builds upon previous knowledge, reinforcing core principles while introducing new complexities. This approach helps students develop a deeper understanding and the ability to tackle more challenging problems involving exponents.

Teacher's Script: Now, let's look at another example that combines multiplication and division with both positive and negative exponents: (x⁶ * x^-10) / x². First, let's multiply the terms in the numerator by adding their exponents: x⁶ * x^-10 = x^-4. Now, we divide this result by x², which means we subtract the exponents: x^-4 / x² = x^{(-4 - 2)} = x^-6. How can we express this in a more familiar form? It's equal to 1/x^6. Notice how the positive and negative exponents interacted throughout the problem, resulting in a negative exponent. This example shows why it's crucial to carefully follow the rules of exponents and keep track of signs!

For a complete collection of math examples related to Exponents click on this link: Math Examples: Laws of Exponents Collection.