Solving Exponents A Step-by-Step Guide To (-3)³ × (-3)⁵ / (-3)¹⁰

Hey guys! Today, let's dive into the fascinating world of exponents and tackle a problem that might seem a bit daunting at first, but trust me, it's super manageable once we break it down. We're going to solve the expression (-3)³ × (-3)⁵ / (-3)¹⁰. Don't worry if exponents feel like a foreign language right now; we'll translate everything into plain English (and math!). Exponents are a fundamental concept in mathematics, and understanding them opens the door to more advanced topics in algebra, calculus, and beyond. This article will not only provide the solution to the given expression but also delve into the underlying principles that govern exponents. By mastering these concepts, you'll be well-equipped to tackle a wide range of mathematical problems involving powers and exponents. We'll start by revisiting the basic definition of exponents and then gradually move towards understanding the rules that govern their manipulation. So, buckle up and let's embark on this mathematical journey together!

Understanding Exponents: The Basics

Before we jump into solving the problem, let's quickly recap what exponents actually mean. An exponent tells you how many times a base number is multiplied by itself. For example, in the expression aⁿ, a is the base, and n is the exponent. This means we multiply a by itself n times. So, 2³ (2 to the power of 3) means 2 × 2 × 2, which equals 8. Similarly, 5² (5 to the power of 2) means 5 × 5, which equals 25. Now, when we deal with negative bases, like in our problem (-3)³, it's crucial to remember the rules for multiplying negative numbers. A negative number multiplied by a negative number gives a positive result, while a positive number multiplied by a negative number gives a negative result. This basic concept is the foundation upon which we build our understanding of exponents. The exponent indicates the number of times the base is multiplied by itself, and this principle applies whether the base is positive, negative, or even a variable. Grasping this foundational idea will make the rules of exponents much easier to understand and apply.

The Product of Powers Rule

One of the key rules we'll use to solve our problem is the product of powers rule. This rule states that when you multiply two exponents with the same base, you simply add the powers. Mathematically, this looks like this: a^m × aⁿ = a^m+n. Why does this work? Let's think about it. If we have 2² × 2³, it means (2 × 2) × (2 × 2 × 2). This is the same as multiplying 2 by itself five times (2⁵). So, we added the exponents (2 + 3 = 5). This rule is a shortcut that saves us from writing out long multiplications. It simplifies the process of multiplying exponential expressions with the same base. The product of powers rule is not just a mathematical trick; it's a direct consequence of the definition of exponents. By understanding the underlying logic, we can confidently apply this rule in various situations. Remember, this rule only applies when the bases are the same. For example, we can't directly apply this rule to expressions like 2² × 3³, because the bases (2 and 3) are different.

The Quotient of Powers Rule

Another essential rule for our problem is the quotient of powers rule. This rule states that when you divide two exponents with the same base, you subtract the powers. In mathematical terms: a^m / aⁿ = a^m-n. Let's see why this works. Imagine we have 3⁵ / 3². This means (3 × 3 × 3 × 3 × 3) / (3 × 3). We can cancel out two 3s from the numerator and denominator, leaving us with 3 × 3 × 3, which is 3³. So, we subtracted the exponents (5 - 2 = 3). The quotient of powers rule is the inverse operation of the product of powers rule. It helps us simplify division problems involving exponents. Just like the product of powers rule, this rule is valid only when the bases are the same. When dealing with complex expressions, it's often helpful to break them down into simpler parts and apply the quotient of powers rule step by step. This rule is a powerful tool for simplifying expressions and solving equations involving exponents.

Solving the Problem: (-3)³ × (-3)⁵ / (-3)¹⁰

Now that we've armed ourselves with the necessary exponent rules, let's tackle our problem: (-3)³ × (-3)⁵ / (-3)¹⁰. The key here is to apply the rules in the correct order. First, we'll use the product of powers rule to simplify the numerator. We have (-3)³ × (-3)⁵. Both terms have the same base (-3), so we can add the exponents: 3 + 5 = 8. This gives us (-3)⁸. So, our expression now looks like this: (-3)⁸ / (-3)¹⁰. Next, we'll apply the quotient of powers rule. We have (-3)⁸ / (-3)¹⁰. Again, the bases are the same, so we subtract the exponents: 8 - 10 = -2. This gives us (-3)⁻². So, the expression simplifies to (-3)⁻². But wait, we're not quite done yet! We have a negative exponent. Negative exponents have a special meaning. They indicate the reciprocal of the base raised to the positive exponent. In other words, a^-n = 1 / aⁿ. Applying this to our problem, (-3)⁻² becomes 1 / (-3)². Finally, we can evaluate (-3)². This means (-3) × (-3), which equals 9. So, our final answer is 1 / 9. See, it wasn't so scary after all! By carefully applying the rules of exponents, we were able to simplify a complex expression and arrive at a clear and concise answer. Remember to take your time, break the problem down into smaller steps, and don't hesitate to review the rules as needed.

Dealing with Negative Exponents

As we saw in the solution, negative exponents might seem a bit strange at first, but they are simply a way of representing reciprocals. Remember, a^-n is the same as 1 / aⁿ. This means that if you have a term with a negative exponent, you can move it to the denominator (or vice versa) and change the sign of the exponent. For example, 2⁻³ is the same as 1 / 2³. And 1 / 5⁻² is the same as 5². Understanding this relationship between negative exponents and reciprocals is crucial for simplifying expressions and solving equations. Negative exponents are not just a mathematical curiosity; they have practical applications in various fields, including science, engineering, and finance. For instance, they are used to represent very small numbers in scientific notation. Mastering negative exponents will significantly enhance your mathematical toolbox and enable you to tackle a wider range of problems.

Final Answer: 1/9

So, after carefully applying the rules of exponents, we've arrived at the solution: (-3)³ × (-3)⁵ / (-3)¹⁰ = 1/9. We started by understanding the basics of exponents, then learned the product of powers and quotient of powers rules, and finally, we tackled negative exponents. This problem is a great example of how understanding the fundamental principles of mathematics can help you solve even seemingly complex problems. Remember, practice makes perfect! The more you work with exponents, the more comfortable you'll become with them. Don't be afraid to make mistakes; they are a valuable part of the learning process. Keep exploring, keep questioning, and most importantly, keep having fun with math! We've journeyed together through the realm of exponents, unraveling the intricacies of powers and their manipulation. With a solid understanding of these fundamental concepts, you're now better equipped to navigate the world of mathematics with confidence and clarity. Congratulations on mastering this important mathematical skill!

Practice Problems

To solidify your understanding, try solving these practice problems:

1. (2⁴ × 2²) / 2⁵
2. (5³ × 5⁻¹) / 5²
3. (-4)² × (-4)³ / (-4)⁴

Good luck, and happy calculating!