Decoding Exponents Solving (81/16)^(3/4) ÷ (32/243)^(-3/5) Of (27/8)^(-2/3)

Aug 5, 2025 by ADMIN 76 views

Unraveling the Mystery of Exponents: A Deep Dive into (81/16)^(3/4) ÷ (32/243)^(-3/5) of (27/8)^(-2/3)

Hey guys! Today, we're going to embark on an exciting mathematical journey to solve a fascinating problem involving exponents and fractions. This isn't just about crunching numbers; it's about understanding the underlying principles and how they all come together. So, buckle up, and let's dive in!

Understanding the Basics of Exponents

Before we tackle the main problem, let's refresh our understanding of exponents. In simple terms, an exponent tells us how many times a number (the base) is multiplied by itself. For example, in the expression 2^3, 2 is the base, and 3 is the exponent. This means we multiply 2 by itself three times: 2 * 2 * 2 = 8. But what happens when we have fractional exponents or negative exponents? That's where things get a little more interesting, and it's crucial for solving our problem. A fractional exponent, such as 1/2, indicates a root. For instance, x^(1/2) is the same as the square root of x, and x^(1/3) is the cube root of x. These roots are essential for simplifying expressions involving fractions raised to fractional powers. On the other hand, a negative exponent indicates a reciprocal. For example, x^(-1) is the same as 1/x. This means we flip the base and change the sign of the exponent. Negative exponents are super handy when dealing with division, as we'll see in our problem. To truly master exponents, it's not enough to just memorize these rules; you need to understand why they work. Think about the relationship between multiplication and exponents. When you multiply two numbers with the same base, you add their exponents (e.g., x^a * x^b = x^(a+b)). This principle extends to fractional and negative exponents as well. By understanding the underlying logic, you can tackle even the most complex exponential expressions with confidence. So, next time you see an exponent, don't just think of it as a number; think of it as a powerful tool for simplifying and solving mathematical problems. Remember, the key is to break down the problem into smaller, manageable steps and apply the rules of exponents systematically. With a little practice, you'll be solving these problems like a pro!

Breaking Down the Expression (81/16)^(3/4)

Let's start by dissecting the first part of our expression: (81/16)^(3/4). This might look intimidating at first, but don't worry, we'll break it down step by step. The key here is to recognize that we're dealing with a fraction raised to a fractional exponent. Remember, a fractional exponent represents both a power and a root. In this case, 3/4 means we need to raise the fraction to the power of 3 and then take the fourth root (or vice versa – the order doesn't matter thanks to the properties of exponents!). The first thing we should do is try to express the base, 81/16, as powers of prime numbers. This will make it easier to apply the exponent. We know that 81 is 3^4 (3 multiplied by itself four times) and 16 is 2^4 (2 multiplied by itself four times). So, we can rewrite 81/16 as (3^4)/(24). Now, we can rewrite the entire expression as ((3^4)/(24))^(3/4). This is where the magic happens! When we have a power raised to another power, we multiply the exponents. So, we have (3⁴⁾(3/4) in the numerator and (2⁴⁾(3/4) in the denominator. Multiplying the exponents, we get 4 * (3/4) = 3. Therefore, the numerator simplifies to 3^3, which is 3 * 3 * 3 = 27. Similarly, in the denominator, we get 2^3, which is 2 * 2 * 2 = 8. So, after simplifying, (81/16)^(3/4) becomes 27/8. See? It's not as scary as it looked! By breaking down the expression and applying the rules of exponents step by step, we were able to simplify it to a much more manageable form. This is a crucial skill for tackling complex mathematical problems. Remember, always look for opportunities to express numbers as powers of prime factors, and don't be afraid to break down the problem into smaller parts. With practice, you'll be able to handle these types of expressions with ease.

Decoding (32/243)^(-3/5)

Next up, we have (32/243)^(-3/5). This one looks a bit trickier because of the negative exponent, but fear not! We know that a negative exponent means we need to take the reciprocal of the base. In other words, we flip the fraction and change the sign of the exponent. So, (32/243)^(-3/5) becomes (243/32)^(3/5). Now, we're dealing with a positive fractional exponent, which we know how to handle. Just like before, the first step is to express the base as powers of prime numbers. We know that 32 is 2^5 (2 multiplied by itself five times) and 243 is 3^5 (3 multiplied by itself five times). So, we can rewrite 243/32 as (3^5)/(25). Now, our expression looks like ((3^5)/(25))^(3/5). Again, we have a power raised to another power, so we multiply the exponents. In the numerator, we have (3⁵⁾(3/5), which simplifies to 3^(5 * (3/5)) = 3^3 = 27. In the denominator, we have (2⁵⁾(3/5), which simplifies to 2^(5 * (3/5)) = 2^3 = 8. Therefore, (243/32)^(3/5) simplifies to 27/8. But wait, we're not done yet! Remember, we started with (32/243)^(-3/5), which we flipped to get (243/32)^(3/5). So, the value of (32/243)^(-3/5) is also 27/8. By using the property of negative exponents to flip the fraction and then applying the rules for fractional exponents, we were able to simplify this seemingly complex expression. The key takeaway here is to not be intimidated by negative exponents. Just remember to flip the base and change the sign of the exponent, and you'll be well on your way to solving the problem.

Unraveling (27/8)^(-2/3)

Let's tackle the last piece of the puzzle: (27/8)^(-2/3). By now, you guys are probably getting the hang of this! We have another negative exponent, so what do we do? We flip the fraction and change the sign of the exponent, of course! So, (27/8)^(-2/3) becomes (8/27)^(2/3). Now, we have a positive fractional exponent. Time to express the base as powers of prime numbers. We know that 8 is 2^3 (2 multiplied by itself three times) and 27 is 3^3 (3 multiplied by itself three times). So, we can rewrite 8/27 as (2^3)/(33). Our expression now looks like ((2^3)/(33))^(2/3). Again, we multiply the exponents. In the numerator, we have (2³⁾(2/3), which simplifies to 2^(3 * (2/3)) = 2^2 = 4. In the denominator, we have (3³⁾(2/3), which simplifies to 3^(3 * (2/3)) = 3^2 = 9. Therefore, (8/27)^(2/3) simplifies to 4/9. So, (27/8)^(-2/3) is equal to 4/9. By consistently applying the same principles – flipping the base for negative exponents and expressing numbers as powers of prime factors – we've simplified another complex expression. The beauty of mathematics is that once you understand the rules, you can apply them to a wide range of problems. This problem demonstrates the power of breaking down complex expressions into smaller, more manageable parts. By focusing on one step at a time and using the properties of exponents, we can conquer even the most challenging mathematical puzzles.

Putting It All Together

Now that we've simplified each part of the expression, it's time to put it all together. Our original problem was: (81/16)^(3/4) ÷ (32/243)^(-3/5) of (27/8)^(-2/3). We've already determined that:

(81/16)^(3/4) = 27/8
(32/243)^(-3/5) = 27/8
(27/8)^(-2/3) = 4/9

So, we can rewrite the problem as: (27/8) ÷ (27/8) of (4/9). Now, remember the order of operations (PEMDAS/BODMAS). The word "of" in this context means multiplication. So, we first need to calculate (27/8) of (4/9), which is (27/8) * (4/9). To multiply fractions, we multiply the numerators and the denominators: (27 * 4) / (8 * 9) = 108/72. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 36. So, 108/72 simplifies to 3/2. Now, our problem looks like: (27/8) ÷ (3/2). To divide fractions, we multiply by the reciprocal of the divisor. In other words, we flip the second fraction and multiply. So, (27/8) ÷ (3/2) becomes (27/8) * (2/3). Multiplying the numerators and the denominators, we get (27 * 2) / (8 * 3) = 54/24. Again, we can simplify this fraction. The greatest common divisor of 54 and 24 is 6. So, dividing both by 6, we get 9/4. Therefore, the final answer to our problem is 9/4. This whole journey shows how important it is to tackle problems systematically. We started with a complex expression, broke it down into smaller parts, simplified each part using the rules of exponents, and then combined the results using the correct order of operations. It's like solving a puzzle – each step builds upon the previous one, leading us to the final solution.

Conclusion: Mastering Exponents

Wow, guys, we've made it! We successfully navigated a challenging problem involving exponents and fractions. We started by understanding the basic rules of exponents, including fractional and negative exponents. Then, we broke down the complex expression into smaller, more manageable parts. We simplified each part by expressing numbers as powers of prime factors and applying the rules of exponents step by step. Finally, we put it all together, using the order of operations to arrive at the final answer. The key takeaways from this journey are:

Understanding the fundamentals: Make sure you have a solid grasp of the rules of exponents, including fractional and negative exponents.
Breaking down complex problems: Don't be intimidated by a large expression. Break it down into smaller parts and tackle each part individually.
Systematic approach: Follow a clear and consistent approach. Express numbers as powers of prime factors, apply the rules of exponents, and use the order of operations.
Practice, practice, practice: The more you practice, the more comfortable you'll become with these types of problems. You'll start to see patterns and develop your own strategies for solving them.

Exponents are a fundamental concept in mathematics, and mastering them will open doors to more advanced topics. So, keep practicing, keep exploring, and keep challenging yourselves. And remember, math can be fun! Until next time, happy calculating!