Simplify A^-3 / B^-2 For A=5 And B=6 A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of negative exponents and fractional expressions. Today, we're tackling a classic problem: simplifying the expression a^-3 / b^-2, where a equals 5 and b equals 6. This might seem a bit daunting at first, but trust me, we'll break it down step by step and make it super easy to understand. We'll not only solve this specific problem but also uncover the underlying principles of negative exponents and how they interact with fractions. So, buckle up and let's get started!

Understanding Negative Exponents

Before we jump into the problem, let's make sure we're all on the same page about negative exponents. Negative exponents can be a bit tricky, but the core concept is actually quite simple. A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. In simpler terms, x^-n is the same as 1 / x^n. This means that instead of multiplying the base by itself a certain number of times, we're dividing 1 by the base raised to that number. For example, 2^-3 is equal to 1 / 2^3, which is 1 / 8. This principle is crucial for simplifying expressions with negative exponents and forms the foundation for our problem today. Understanding this concept allows us to transform expressions with negative exponents into fractions, making them easier to manipulate and solve. Remember, the negative sign in the exponent doesn't mean the result is negative; it simply indicates a reciprocal.

Now, let's consider why this works. Think about the pattern of exponents: x^3 = x * x * x, x^2 = x * x, x^1 = x. If we continue this pattern downwards, x^0 should be 1 (since we're essentially dividing by x each time). Then, x^-1 would be 1 / x, x^-2 would be 1 / (x * x), and so on. This pattern visually demonstrates why a negative exponent results in a reciprocal. This understanding will be vital as we move forward and apply it to our specific problem. We will be using this understanding of negative exponents to transform our initial expression into a more manageable form, which will then allow us to substitute the given values of 'a' and 'b' and arrive at the final answer. By mastering this concept, you'll be well-equipped to tackle a wide range of exponent-related problems.

Breaking Down the Expression: a^-3 / b^-2

Now that we've got a solid grasp of negative exponents, let's tackle the expression a^-3 / b^-2. Remember, our goal is to simplify this expression before we plug in the values for a and b. Using our understanding of negative exponents, we can rewrite a^-3 as 1 / a^3 and b^-2 as 1 / b^2. This transformation is the key to making the expression more manageable. So, our expression now looks like (1 / a^3) / (1 / b^2). But how do we deal with a fraction divided by another fraction? This is where another fundamental math principle comes into play: dividing by a fraction is the same as multiplying by its reciprocal. This is a crucial rule to remember when dealing with complex fractions. It allows us to convert a division problem into a multiplication problem, which is often easier to handle.

So, we can rewrite (1 / a^3) / (1 / b^2) as (1 / a^3) * (b^2 / 1). This simplifies to b^2 / a^3. See how much cleaner that looks? We've successfully transformed the expression with negative exponents and a complex fraction into a simple fraction with positive exponents. This is a significant step forward. By applying these fundamental rules of exponents and fractions, we've transformed the original expression into a much more manageable form. The ability to manipulate expressions in this way is a cornerstone of algebraic problem-solving. The key takeaway here is that by understanding and applying the rules of exponents and fractions, we can simplify complex expressions and make them easier to work with. Now, we're ready for the final step: substituting the values of a and b.

Substituting Values and Finding the Solution

Alright, we've simplified our expression to b^2 / a^3. Now comes the fun part: plugging in the values! We know that a = 5 and b = 6. So, let's substitute those values into our simplified expression. This gives us 6^2 / 5^3. Now, we just need to calculate these powers. 6^2 means 6 multiplied by itself, which is 36. And 5^3 means 5 multiplied by itself three times, which is 5 * 5 * 5 = 125. So, our expression becomes 36 / 125. Can we simplify this fraction further? Let's check for common factors. 36 has factors like 2, 3, 4, 6, 9, 12, and 18. 125 has factors like 5 and 25. Since they don't share any common factors other than 1, the fraction 36 / 125 is already in its simplest form.

Therefore, the final answer to our problem, a^-3 / b^-2 with a = 5 and b = 6, is 36 / 125. We've successfully navigated the world of negative exponents and fractions to arrive at our solution! This entire process highlights the importance of breaking down complex problems into smaller, manageable steps. By understanding the underlying principles and applying them systematically, we can solve even seemingly challenging problems. We started with a potentially intimidating expression with negative exponents, simplified it using the rules of exponents and fractions, and then substituted the given values to find our final answer. This is a testament to the power of a step-by-step approach and a solid understanding of mathematical concepts. Remember, practice makes perfect, so keep exploring and tackling new problems!

Key Takeaways and Further Exploration

So, what have we learned today? We've not only solved a specific problem involving negative exponents and fractions, but we've also reinforced some crucial mathematical concepts. We've seen how negative exponents indicate reciprocals, how dividing by a fraction is the same as multiplying by its reciprocal, and how to simplify expressions step-by-step. These are fundamental skills that will serve you well in your mathematical journey. But the learning doesn't stop here! There's always more to explore. For instance, you could try tackling more complex expressions with multiple negative exponents, or expressions involving variables and other mathematical operations. You could also delve deeper into the properties of exponents and how they interact with different types of numbers, like radicals or complex numbers. The key is to keep practicing and keep challenging yourself.

Consider exploring problems with nested exponents, such as (a^-2)-3, or expressions involving multiple variables. You can also investigate how these concepts apply in different areas of mathematics, such as algebra, calculus, and even physics. For example, negative exponents are often used to represent very small numbers in scientific notation. Understanding these concepts will not only improve your problem-solving skills but also give you a deeper appreciation for the elegance and interconnectedness of mathematics. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying principles and applying them creatively. So, keep exploring, keep experimenting, and most importantly, keep having fun with math! You've got this!

Simplify the expression a^-3 / b^-2 when a = 5 and b = 6.

Simplify a^-3 / b^-2 for a=5 and b=6: A Step-by-Step Guide