Equivalent Expression For (3/7)^2 * (2/3)^-3)^-1 A Math Guide

Hey guys! Ever find yourself staring at a math problem that looks like it's written in another language? Well, you're not alone! Let's break down one of those seemingly complex expressions today. We're going to dissect an expression involving fractions, exponents, and a negative power, step by step, so that by the end, you'll feel like a total math whiz. Our mission? To figure out which expression is equivalent to this beast: $\left(\left(\frac{3}{7}\right)^2 \cdot\left(\frac{2}{3}\right)^{-3}\right)^{-1}$. Buckle up, because we're about to dive deep into the world of exponents and fractions!

Understanding the Basics: Exponents and Fractions

Before we even think about tackling the main problem, let's make sure we're all on the same page with some fundamental concepts. Exponents are a shorthand way of showing repeated multiplication. For example, $2^3$ (read as “2 to the power of 3”) means 2 multiplied by itself three times: $2 \cdot 2 \cdot 2 = 8$. The number being multiplied (in this case, 2) is called the base, and the little number up top (in this case, 3) is the exponent or power.

Now, let's throw fractions into the mix. A fraction represents a part of a whole. It has a numerator (the top number) and a denominator (the bottom number). When we raise a fraction to a power, we're essentially raising both the numerator and the denominator to that power. So, $\left(\frac{a}{b}\right)^n$ is the same as $\frac{a^n}{b^n}$. This is a crucial rule to remember!

But wait, there's more! What happens when we encounter a negative exponent? This is where things get a little bit funky, but don't worry, it's not as scary as it looks. A negative exponent tells us to take the reciprocal of the base and then raise it to the positive version of the exponent. In other words, $a^{-n}$ is the same as $\frac{1}{a^n}$. Similarly, for a fraction, $\left(\frac{a}{b}\right)^{-n}$ is the same as $\left(\frac{b}{a}\right)^n$. We essentially flip the fraction and change the sign of the exponent. This reciprocal property is going to be our best friend in solving the problem at hand.

Diving Deeper into Negative Exponents with Fractions

Let's really nail down this concept of negative exponents with fractions because it's super important for this problem. Think of a negative exponent as an invitation to flip things around. When you see $\left(\frac{2}{3}\right)^{-3}$, your brain should immediately think, “Okay, I'm going to flip this fraction and make the exponent positive.” That means $\left(\frac{2}{3}\right)^{-3}$ becomes $\left(\frac{3}{2}\right)^3$. Now, we're dealing with a positive exponent, which we already know how to handle.

We can then apply the rule we discussed earlier: $\left(\frac{3}{2}\right)^3$ is the same as $\frac{3^3}{2^3}$, which is $\frac{27}{8}$. See? Not so intimidating after all! The key is to break it down step by step and remember the rules.

This idea of flipping the fraction is essentially the same as taking the reciprocal. The reciprocal of a number is simply 1 divided by that number. So, the reciprocal of $\frac{2}{3}$ is $\frac{1}{\frac{2}{3}}$, which simplifies to $\frac{3}{2}$. This connection between negative exponents and reciprocals is what makes this whole process work.

Understanding this relationship is not just about memorizing a rule; it's about grasping the underlying concept. When you understand why a rule works, you're much more likely to remember it and apply it correctly in different situations. So, take a moment to really think about what's happening when you have a negative exponent with a fraction. Visualize the flipping action, and connect it to the idea of reciprocals. Trust me, it will make your life a whole lot easier when you're tackling complex math problems!

Breaking Down the Main Expression

Alright, now that we've refreshed our understanding of exponents and fractions, let's circle back to the original problem: $\left(\left(\frac{3}{7}\right)^2 \cdot\left(\frac{2}{3}\right)^{-3}\right)^{-1}$. This might look a bit scary, but we're going to take it one step at a time, just like eating an elephant – one bite at a time, right? The key here is to follow the order of operations (PEMDAS/BODMAS) and apply the exponent rules we just discussed.

The first thing we need to tackle is the inner part of the expression: $\left(\frac{3}{7}\right)^2 \cdot\left(\frac{2}{3}\right)^{-3}$. Notice that we have two terms multiplied together, each with its own exponent. Let's deal with each term separately.

For the first term, $\left(\frac{3}{7}\right)^2$, we simply apply the rule of raising a fraction to a power: $\left(\frac{3}{7}\right)^2 = \frac{3^2}{7^2} = \frac{9}{49}$. So far, so good!

Now, let's tackle the second term, $\left(\frac{2}{3}\right)^{-3}$. Ah, a negative exponent! Remember what we learned? We flip the fraction and change the sign of the exponent: $\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3$. Now we have a positive exponent, which we can easily handle: $\left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}$. Excellent!

Now we can substitute these simplified terms back into our inner expression: $\frac{9}{49} \cdot \frac{27}{8}$. To multiply fractions, we simply multiply the numerators and the denominators: $\frac{9}{49} \cdot \frac{27}{8} = \frac{9 \cdot 27}{49 \cdot 8} = \frac{243}{392}$. Phew! We've simplified the inner expression quite a bit.

The Outer Exponent: Taking it Home

We're not done yet, though! We still have that outer exponent of -1 to deal with. Remember, the entire expression we just simplified, $\frac{243}{392}$, is being raised to the power of -1: $\left(\frac{243}{392}\right)^{-1}$.

This is where our friend, the negative exponent, comes to the rescue again! We know that raising something to the power of -1 means taking its reciprocal. So, $\left(\frac{243}{392}\right)^{-1}$ is simply the reciprocal of $\frac{243}{392}$, which is $\frac{392}{243}$.

And there you have it! We've successfully navigated the maze of exponents and fractions, and we've arrived at our final simplified expression: $\frac{392}{243}$. This is the equivalent expression for the original problem. Woohoo! High fives all around!

Alternative Approaches and Key Takeaways

Now, you might be wondering, “Are there other ways to solve this problem?” Absolutely! In mathematics, there's often more than one path to the solution. One alternative approach would be to distribute the outer exponent of -1 at the very beginning. Remember the rule that $(a \cdot b)^n = a^n \cdot b^n$? We can apply a similar concept here.

If we distribute the -1, we get: $\left(\left(\frac{3}{7}\right)^2 \cdot\left(\frac{2}{3}\right)^{-3}\right)^{-1} = \left(\frac{3}{7}\right)^{2 \cdot -1} \cdot \left(\frac{2}{3}\right)^{-3 \cdot -1} = \left(\frac{3}{7}\right)^{-2} \cdot \left(\frac{2}{3}\right)^{3}$.

Now we have two terms with exponents. We can apply the negative exponent rule to the first term: $\left(\frac{3}{7}\right)^{-2} = \left(\frac{7}{3}\right)^2 = \frac{49}{9}$. The second term is already in a friendly form: $\left(\frac{2}{3}\right)^{3} = \frac{8}{27}$.

Multiplying these together, we get: $\frac{49}{9} \cdot \frac{8}{27} = \frac{49 \cdot 8}{9 \cdot 27} = \frac{392}{243}$.

Notice that we arrived at the same answer, $\frac{392}{243}$, using a different method! This highlights a crucial point in mathematics: there's often more than one way to skin a cat (or solve a math problem!). The key is to understand the rules and apply them strategically.

Key Takeaways and Pro Tips

So, what are the key takeaways from this mathematical adventure? First and foremost, understanding the rules of exponents, especially negative exponents, is crucial. Remember that a negative exponent means taking the reciprocal. Second, breaking down complex expressions into smaller, manageable steps is essential. Don't try to do everything at once! Third, remember that there's often more than one way to solve a problem. Explore different approaches and find what works best for you. Finally, practice makes perfect! The more you work with exponents and fractions, the more comfortable you'll become.

Here are a few pro tips to keep in mind:

• Write out each step: Don't try to skip steps in your head, especially when you're first learning. Writing things out helps you keep track of your work and reduces the chance of making mistakes.
• Double-check your work: It's always a good idea to go back and check your steps, especially if you're working on a test or an important assignment.
• Use parentheses: Parentheses can help you keep track of what operations you're doing and in what order.
• Don't be afraid to ask for help: If you're stuck, don't hesitate to ask your teacher, a tutor, or a friend for help. We all get stuck sometimes!

By mastering these concepts and practicing regularly, you'll be able to tackle even the most intimidating expressions with confidence. So, keep practicing, keep exploring, and keep having fun with math! You got this!

Practice Problems to Sharpen Your Skills

To really solidify your understanding, let's try a few practice problems. These will give you a chance to apply the concepts we've discussed and build your confidence. Remember, the key is to break down each problem into smaller steps and apply the rules of exponents and fractions carefully.

Problem 1: Simplify the expression $\left(\left(\frac{5}{2}\right)^{-2} \cdot \left(\frac{1}{3}\right)^{2}\right)^{-1}$.

Problem 2: Find an equivalent expression for $\left(\frac{4}{9}\right)^{-3} \div \left(\frac{2}{3}\right)^{-2}$.

Problem 3: Evaluate $\left(\left(\frac{1}{2}\right)^3 \cdot \left(\frac{4}{5}\right)^{-1}\right)^{-2}$.

Take your time, work through each step, and remember the strategies we discussed. Don't be afraid to make mistakes – that's how we learn! Once you've solved these problems, you'll be well on your way to becoming an exponent and fraction master.

Solutions and Explanations

Okay, guys, let's check our work and make sure we're on the right track! Here are the solutions to the practice problems, along with detailed explanations to help you understand each step.

Solution 1: $\left(\left(\frac{5}{2}\right)^{-2} \cdot \left(\frac{1}{3}\right)^{2}\right)^{-1}$

• First, let's simplify the inner exponents: $\left(\frac{5}{2}\right)^{-2} = \left(\frac{2}{5}\right)^{2} = \frac{4}{25}$ and $\left(\frac{1}{3}\right)^{2} = \frac{1}{9}$.
• Now, multiply the simplified terms: $\frac{4}{25} \cdot \frac{1}{9} = \frac{4}{225}$.
• Finally, apply the outer exponent of -1: $\left(\frac{4}{225}\right)^{-1} = \frac{225}{4}$.

So, the simplified expression is $\frac{225}{4}$.

Solution 2: $\left(\frac{4}{9}\right)^{-3} \div \left(\frac{2}{3}\right)^{-2}$

• Let's deal with the negative exponents first: $\left(\frac{4}{9}\right)^{-3} = \left(\frac{9}{4}\right)^{3} = \frac{729}{64}$ and $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}$.
• Now, remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we have $\frac{729}{64} \div \frac{9}{4} = \frac{729}{64} \cdot \frac{4}{9}$.
• Multiply the fractions: $\frac{729}{64} \cdot \frac{4}{9} = \frac{2916}{576}$.
• Simplify the fraction: $\frac{2916}{576} = \frac{81}{16}$.

Therefore, the equivalent expression is $\frac{81}{16}$.

Solution 3: $\left(\left(\frac{1}{2}\right)^3 \cdot \left(\frac{4}{5}\right)^{-1}\right)^{-2}$

• Simplify the inner exponents: $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$ and $\left(\frac{4}{5}\right)^{-1} = \frac{5}{4}$.
• Multiply the simplified terms: $\frac{1}{8} \cdot \frac{5}{4} = \frac{5}{32}$.
• Apply the outer exponent of -2: $\left(\frac{5}{32}\right)^{-2} = \left(\frac{32}{5}\right)^{2} = \frac{1024}{25}$.

Thus, the evaluated expression is $\frac{1024}{25}$.

How did you do? If you got all the answers correct, congratulations! You're well on your way to mastering exponents and fractions. If you missed a few, don't worry! Go back and review the steps, and try to identify where you went wrong. Remember, practice and perseverance are the keys to success in mathematics.

Final Thoughts and Encouragement

We've covered a lot of ground in this article, from the basic rules of exponents and fractions to tackling complex expressions and alternative solution methods. We've also worked through some practice problems and discussed common mistakes and strategies for success.

The most important thing to remember is that mathematics is not about memorizing formulas; it's about understanding concepts and developing problem-solving skills. When you approach a problem, don't just try to apply a formula blindly. Take the time to understand what the problem is asking and why the rules work the way they do.

And remember, everyone struggles with math sometimes. Don't get discouraged if you don't understand something right away. Keep practicing, keep asking questions, and keep exploring. The more you engage with mathematics, the more confident and skilled you'll become.

So, go forth and conquer those exponents and fractions! You've got the tools and the knowledge you need to succeed. And remember, math can be fun! Embrace the challenge, enjoy the process, and celebrate your successes along the way. You're doing great!