Simplifying (15×17)⁵×4⁷ / (30×34)⁴ A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like a monster but is actually a kitten in disguise? Today, we're going to tackle one such beast and break it down into purr-fectly manageable steps. Our mission: simplifying the expression (15×17)⁵×4⁷ / (30×34)⁴. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Problem

Before we dive into the nitty-gritty, let's make sure we're all on the same page. The problem presents us with a complex fraction involving exponents and multiplication. Simplifying it means reducing it to its simplest form, where there are no common factors between the numerator and the denominator. Think of it like decluttering your room – we want to get rid of anything unnecessary and leave only the essentials. In this case, the essentials are the prime factors and their lowest possible exponents.

The expression we're dealing with is (15×17)⁵×4⁷ / (30×34)⁴. At first glance, it might seem intimidating, but don't worry! We're going to break it down piece by piece. Remember, the key to simplifying complex expressions is to identify common factors and use the rules of exponents wisely. Exponents are just a shorthand way of writing repeated multiplication, and they have some cool properties that we can exploit. For example, (a × b)ⁿ = aⁿ × bⁿ and aⁿ / aᵐ = aⁿ⁻ᵐ. These rules will be our best friends in this journey. So, let's not be scared by the size of the numbers or the exponents. Instead, let's put on our detective hats and start looking for clues – those hidden factors that will help us unravel this mathematical mystery.

Breaking Down the Numerator

First, let's focus on the numerator: (15×17)⁵×4⁷. The numerator is the top part of our fraction, and it's where we'll start our simplification journey. To simplify this, we need to break down each number into its prime factors. Prime factors are the smallest building blocks of a number – prime numbers that multiply together to give you the original number. For example, the prime factors of 12 are 2 × 2 × 3. So, let's apply this concept to our numerator. We have 15, 17, and 4. 15 can be broken down into 3 × 5. 17 is already a prime number, so we can leave it as it is. And 4 can be broken down into 2 × 2, or 2². Now, let's rewrite the numerator using these prime factors: (3 × 5 × 17)⁵ × (2²)⁷. Remember that rule we talked about, (a × b)ⁿ = aⁿ × bⁿ? We can use it here to distribute the exponent of 5 across the terms inside the parentheses: 3⁵ × 5⁵ × 17⁵ × (2²)⁷. Now, let's tackle that last term, (2²)⁷. Another handy rule of exponents comes into play here: (aⁿ)ᵐ = aⁿᵐ. So, (2²)⁷ becomes 2¹⁴. Putting it all together, our simplified numerator looks like this: 3⁵ × 5⁵ × 17⁵ × 2¹⁴. Not so scary anymore, right? We've successfully broken down the numerator into its prime factors with their respective exponents. This is a crucial step because it allows us to easily identify common factors with the denominator later on. Remember, the goal is to find those common factors and cancel them out, simplifying the expression as much as possible. So, with the numerator neatly organized, let's turn our attention to the denominator and see what secrets it holds.

Deconstructing the Denominator

Now, let's tackle the denominator: (30×34)⁴. Just like we did with the numerator, our goal here is to break down each number into its prime factors. This will allow us to identify any common factors between the numerator and the denominator, which we can then cancel out to simplify the expression. We have 30 and 34. Let's start with 30. We can break it down into 2 × 15, and then further break down 15 into 3 × 5. So, the prime factorization of 30 is 2 × 3 × 5. Next, let's look at 34. We can break it down into 2 × 17. So, the prime factorization of 34 is 2 × 17. Now, let's rewrite the denominator using these prime factors: (2 × 3 × 5 × 2 × 17)⁴. Notice that we have two 2s in there. We can combine them as 2². So, the expression becomes (2² × 3 × 5 × 17)⁴. Now, we need to distribute the exponent of 4 across all the terms inside the parentheses. Remember our handy rule, (a × b)ⁿ = aⁿ × bⁿ? Let's apply it: (2²)⁴ × 3⁴ × 5⁴ × 17⁴. And we're not done yet! We still need to simplify that (2²)⁴ term. Using the rule (aⁿ)ᵐ = aⁿᵐ, we know that (2²)⁴ becomes 2⁸. So, our fully simplified denominator is: 2⁸ × 3⁴ × 5⁴ × 17⁴. Just like with the numerator, we've successfully broken down the denominator into its prime factors with their respective exponents. This makes it super easy to compare it with the numerator and spot those common factors that we can cancel out. We're getting closer to simplifying the whole expression! With both the numerator and the denominator neatly factored, the next step is the most exciting one: the grand cancellation!

Simplifying the Fraction

Alright, guys, this is where the magic happens! We've broken down both the numerator and the denominator into their prime factors, and now we're ready to simplify the entire fraction. Remember, simplification is all about finding common factors in the numerator and denominator and cancelling them out. It's like simplifying a fraction like 6/8 by dividing both the top and bottom by 2 to get 3/4. We're going to do the same thing here, but with our prime factors and exponents. Our fraction now looks like this:

(3⁵ × 5⁵ × 17⁵ × 2¹⁴) / (2⁸ × 3⁴ × 5⁴ × 17⁴)

Let's start by looking for common factors. We see that we have 3 in both the numerator and the denominator. We have 5 in both, 17 in both, and 2 in both! That's fantastic! Now, we need to figure out how many of each factor we can cancel out. Remember the rule aⁿ / aᵐ = aⁿ⁻ᵐ? This is our weapon of choice for simplifying exponents. Let's apply it to each common factor:

• For 3: We have 3⁵ in the numerator and 3⁴ in the denominator. So, 3⁵ / 3⁴ = 3⁵⁻⁴ = 3¹ = 3
• For 5: We have 5⁵ in the numerator and 5⁴ in the denominator. So, 5⁵ / 5⁴ = 5⁵⁻⁴ = 5¹ = 5
• For 17: We have 17⁵ in the numerator and 17⁴ in the denominator. So, 17⁵ / 17⁴ = 17⁵⁻⁴ = 17¹ = 17
• For 2: We have 2¹⁴ in the numerator and 2⁸ in the denominator. So, 2¹⁴ / 2⁸ = 2¹⁴⁻⁸ = 2⁶

Now, let's put it all together. After cancelling out the common factors, we're left with:

3 × 5 × 17 × 2⁶

This is our simplified expression! We've successfully navigated the maze of exponents and prime factors to arrive at a much cleaner and more manageable form. But we're not quite done yet. Let's calculate the final value to get our ultimate answer.

Calculating the Final Answer

We've simplified our expression to 3 × 5 × 17 × 2⁶. Now, it's time to crunch the numbers and get our final answer. This step is straightforward – we just need to perform the multiplications. Remember, 2⁶ means 2 multiplied by itself six times: 2 × 2 × 2 × 2 × 2 × 2 = 64. So, we have:

3 × 5 × 17 × 64

Let's multiply it step by step:

• 3 × 5 = 15
• 15 × 17 = 255
• 255 × 64 = 16320

So, the final simplified answer is 16320. Wow! We started with a seemingly complex expression and, through careful factorization and simplification, we arrived at a single, neat number. That's the power of mathematics, guys! It can take something that looks intimidating and break it down into manageable, understandable pieces. We've not only solved the problem but also reinforced some important mathematical principles along the way. Remember, the key to simplifying expressions like this is to break them down into their prime factors, use the rules of exponents, and systematically cancel out common factors. With a little practice, you'll be simplifying even the most daunting expressions like a pro.

Conclusion

So, there you have it! We've successfully simplified the expression (15×17)⁵×4⁷ / (30×34)⁴ and arrived at the answer: 16320. We did it by breaking down the numerator and denominator into their prime factors, applying the rules of exponents, cancelling out common factors, and finally, performing the multiplication. This journey might have seemed challenging at first, but by taking it step by step, we've shown that even complex mathematical problems can be tamed. Remember, mathematics is not about memorizing formulas, but about understanding the underlying principles and applying them logically. The skills we've used today – prime factorization, exponent rules, and simplification techniques – are fundamental in mathematics and will serve you well in many other areas. So, keep practicing, keep exploring, and never be afraid to tackle a challenging problem. You might just surprise yourself with what you can achieve!

I hope this step-by-step guide has been helpful and has demystified the process of simplifying complex expressions. Keep practicing, and you'll become a math whiz in no time! And remember, if you ever get stuck, just break it down, piece by piece, and you'll find your way through. Happy simplifying, guys!