Simplify (a^-7b^-8)/(a^5b^5): Exponent Rules Explained

by Mei Lin 55 views

Hey guys! Today, we're diving deep into the fascinating world of algebraic expressions, specifically focusing on how to simplify expressions involving negative exponents. You know, those tricky little exponents that can sometimes make things look a bit complicated? Well, fear not! We're here to break it all down in a way that's super easy to understand. We'll start with a basic problem and then explore the underlying principles and rules that govern how negative exponents behave. By the end of this guide, you'll be a pro at simplifying even the most daunting expressions. This comprehensive guide aims to simplify expressions with negative exponents, a common task in algebra. We’ll explore the rules governing exponents and demonstrate how to apply them effectively. Whether you're a student tackling homework or just brushing up on your math skills, this article will provide a clear, step-by-step approach to mastering this concept. So, let's jump right in and make those negative exponents disappear!

Let's start with the expression we want to simplify:

a−7b−8a5b5\frac{a^{-7} b^{-8}}{a^5 b^5}

At first glance, it might look a bit intimidating with all those negative exponents. But trust me, it's not as scary as it seems! Our goal is to rewrite this expression in a simpler form, ideally without any negative exponents. To do this, we'll need to understand the fundamental rules of exponents and how they work, especially when dealing with negative powers. Think of it like translating a foreign language – once you know the rules, you can decipher anything. We’ll take it step by step, making sure each part is crystal clear. This expression involves variables a{a} and b{b} raised to both positive and negative powers. The key to simplifying it lies in understanding how to handle these exponents. We'll go over the rules for dividing powers with the same base and how negative exponents can be converted to positive ones. So, get ready to simplify this expression and many others like it!

Before we dive into the solution, let's make sure we're all on the same page about what negative exponents actually mean. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. For example, x−n{x^{-n}} is the same as 1xn{\frac{1}{x^n}}. This is a crucial concept to grasp because it's the foundation for simplifying expressions like the one we have. Basically, a negative exponent tells us to move the base to the opposite side of the fraction bar. If it's in the numerator, it moves to the denominator, and vice versa. Once it moves, the exponent becomes positive. It’s like a little mathematical dance! Understanding this principle makes the entire process of simplifying expressions significantly easier. Think of negative exponents as mathematical signals that tell you to flip things around. This simple rule is the key to unlocking the solution to our problem and many more. Now, let’s see how this applies to our specific expression.

To simplify the expression $ rac{a^{-7} b{-8}}{a5 b^5}$, we need to recall the fundamental rules of exponents. These rules are the backbone of simplifying any exponential expression, and mastering them will make your life in algebra so much easier. Here are the two main rules we'll be using:

  1. Quotient Rule: When dividing powers with the same base, you subtract the exponents. Mathematically, this is expressed as: $\frac{xm}{xn} = x^{m-n}$. This rule allows us to combine terms with the same base that are being divided. It's like saying,