Simplifying Exponential Expressions: A Step-by-Step Guide

Hey guys! Let's dive into simplifying this exponential expression together. We've got a cool problem here: $\frac{3a^{-4}}{2b^{-5}} imes 2^3 imes 3^{-1}$. It might look intimidating at first, but don't worry, we'll break it down step by step so it’s super easy to understand. We'll cover everything from understanding negative exponents to combining like terms. By the end of this, you'll be a pro at simplifying these kinds of expressions!

Understanding the Basics of Exponents

Before we jump into the problem, let's quickly recap what exponents are all about. You probably already know this, but a quick refresher never hurts, right? An exponent tells you how many times to multiply a number (the base) by itself. For example, $2^3$ means 2 multiplied by itself three times: $2 imes 2 imes 2 = 8$. Simple enough, yeah?

Now, what about negative exponents? This is where things can get a little tricky, but trust me, it’s not as scary as it looks. A negative exponent simply means you need to take the reciprocal of the base raised to the positive exponent. In other words, $a^{-n}$ is the same as $\frac{1}{a^n}$. So, if we have $a^{-4}$, it's really $\frac{1}{a^4}$. Similarly, $b^{-5}$ is $\frac{1}{b^5}$. But here’s the cool part: when a term with a negative exponent is in the denominator, like in our problem, we can move it to the numerator and make the exponent positive! This is a super handy trick that we’ll use in our simplification.

Another key concept is the product of powers. When you multiply terms with the same base, you add their exponents. For example, $a^m imes a^n = a^{m+n}$. Conversely, when dividing terms with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. These rules are essential for simplifying more complex expressions, so make sure you've got them down. We'll use these rules in the problem we're tackling today, so it's good to have them fresh in your mind. Got it? Great! Let's move on and see how we can apply these rules to simplify the given expression.

Step-by-Step Simplification

Okay, let’s get to the fun part – simplifying the expression $\frac{3a^{-4}}{2b^{-5}} imes 2^3 imes 3^{-1}$. Grab your pencils and let's break it down together!

1. Rewrite Negative Exponents

Our first step is to deal with those negative exponents. Remember, $a^{-4}$ is the same as $\frac{1}{a^4}$, and $b^{-5}$ in the denominator becomes $b^5$ in the numerator. So, let's rewrite the expression:

$\frac{3a^{-4}}{2b^{-5}} imes 2^3 imes 3^{-1} = \frac{3 imes \frac{1}{a^4}}{2 imes \frac{1}{b^5}} imes 2^3 imes \frac{1}{3}$

Now, let’s move that $b^{-5}$ from the denominator up to the numerator, which changes the sign of the exponent:

$= \frac{3b^5}{2a^4} imes 2^3 imes 3^{-1}$

See? We've already made some good progress. The expression looks a little cleaner now, right? No more negative exponents in the fractions, which is always a good start. This step is crucial because it helps us rearrange the terms and combine like terms more easily. Make sure you're comfortable with moving terms between the numerator and denominator – it's a super useful trick!

2. Expand and Rearrange Terms

Next up, let's expand $2^3$ and rewrite $3^{-1}$. We know that $2^3 = 2 imes 2 imes 2 = 8$, and $3^{-1} = \frac{1}{3}$. So, our expression becomes:

$= \frac{3b^5}{2a^4} imes 8 imes \frac{1}{3}$

Now, let's rearrange the terms to group the constants together. This will make it easier to simplify the numerical parts of the expression:

$= \frac{3 imes 8 imes b^5}{2 imes 3 imes a^4}$

Rearranging terms is a simple but powerful technique. It helps us see the structure of the expression more clearly and identify opportunities for simplification. In this case, we've grouped the numbers together, which will allow us to easily cancel out common factors in the next step. Always be on the lookout for ways to rearrange and regroup terms – it can make a big difference in simplifying complex expressions!

3. Simplify Constants

Now comes the satisfying part – simplifying the constants. We have $\frac{3 imes 8}{2 imes 3}$. Notice that we have a 3 in both the numerator and the denominator, so we can cancel those out. Also, 8 divided by 2 is 4. Let's do it:

$= \frac{\cancel{3} imes 8}{2 imes \cancel{3}} imes \frac{b^5}{a^4}$

$= \frac{8}{2} imes \frac{b^5}{a^4}$

$= 4 \times \frac{b^5}{a^4}$

So, our expression simplifies to:

$= \frac{4b^5}{a^4}$

Isn't it cool how much simpler it looks now? Simplifying constants is all about finding common factors and canceling them out. This is a fundamental skill in algebra, and it's something you'll use all the time. Keep an eye out for opportunities to simplify constants – it can drastically reduce the complexity of your expressions and make them much easier to work with.

Final Answer and Wrap-up

Alright, we’ve made it to the end! The simplified form of the expression $\frac{3a^{-4}}{2b^{-5}} imes 2^3 imes 3^{-1}$ is $\frac{4b^5}{a^4}$. Woohoo! You nailed it!

So, the correct answer is A. $\frac{4b^5}{a^4}$.

Key Takeaways

Let's recap what we've learned in this simplification journey:

• Negative Exponents: Remember, $a^{-n} = \frac{1}{a^n}$. Move terms with negative exponents across the fraction bar to make the exponent positive.
• Expanding and Rearranging: Expand terms like $2^3$ and rearrange the expression to group like terms together. This makes it easier to see what can be simplified.
• Simplifying Constants: Look for common factors in the numerator and denominator to cancel out.

These steps are super useful for tackling any exponential expression. Practice them, and you'll become a simplification master in no time! Remember, math is like a puzzle – each piece fits together to reveal the solution. Keep practicing, and you'll get better and better at solving these puzzles. You've got this!

If you have any questions or want to try another problem, just let me know. Keep up the awesome work, and I'll see you in the next math adventure!