Evaluate $\frac{a^{-4}}{b^{-2}}$ When A=3 And B=2

Let's break down how to evaluate the expression $\frac{a^{-4}}{b^{-2}}$ when $a = 3$ and $b = 2$. This involves understanding negative exponents and how to manipulate them. So, grab your calculators (or your mental math muscles) and let’s dive in!

Understanding Negative Exponents

Before we plug in the values, let's make sure we're all on the same page about negative exponents. A negative exponent basically means you take the reciprocal of the base raised to the positive version of that exponent. In other words:

$x^{-n} = \frac{1}{x^n}$

For example, $2^{-1} = \frac{1}{2}$ and $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$. Understanding this principle is absolutely crucial for solving this problem. Without it, you might end up with the wrong answer, and we definitely don't want that, right? It’s like trying to bake a cake without knowing what flour does – you'll probably end up with a mess! So, remember, negative exponents indicate reciprocals.

Also keep in mind that when you have a fraction with negative exponents, like $\frac{a^{-4}}{b^{-2}}$, you can move the terms with negative exponents to the opposite side of the fraction bar and change the sign of the exponent. This is because dividing by a reciprocal is the same as multiplying by the original number. For instance:

$\frac{1}{x^{-n}} = x^n$

Think of it like this: if you're dividing by something really small (like $\frac{1}{9}$), it's the same as multiplying by its inverse (9). This trick is super handy for simplifying expressions and making them easier to work with.

Mastering negative exponents is not just about memorizing rules; it's about understanding the underlying concept. Once you get the hang of it, you'll find that many problems involving exponents become much more manageable. Plus, it's a fundamental concept that pops up in various areas of mathematics and even in fields like physics and engineering. So, invest the time to truly understand it, and you'll reap the benefits down the road.

Substituting the Values

Now that we've refreshed our memory on negative exponents, let's substitute $a = 3$ and $b = 2$ into the expression $\frac{a^{-4}}{b^{-2}}$.

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

Now we apply the rule for negative exponents:

$\frac{3^{-4}}{2^{-2}} = \frac{\frac{1}{3^4}}{\frac{1}{2^2}}$

Simplifying the Expression

Okay, let's simplify those exponents. We know that $3^4 = 3 \times 3 \times 3 \times 3 = 81$ and $2^2 = 2 \times 2 = 4$. So our expression becomes:

$\frac{\frac{1}{81}}{\frac{1}{4}}$

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore:

$\frac{\frac{1}{81}}{\frac{1}{4}} = \frac{1}{81} \times \frac{4}{1} = \frac{4}{81}$

So, the value of the expression $\frac{a^{-4}}{b^{-2}}$ when $a = 3$ and $b = 2$ is $\frac{4}{81}$.

Alternative Approach: Moving Negative Exponents

Another way to solve this, which some people find easier, is to move the terms with negative exponents to the opposite side of the fraction:

$\frac{a^{-4}}{b^{-2}} = \frac{b^2}{a^4}$

Then substitute the values:

$\frac{b^2}{a^4} = \frac{2^2}{3^4} = \frac{4}{81}$

Same answer, different route! Choose the method that clicks best for you.

Why Other Options are Incorrect

Let's briefly look at why the other options are wrong. This can help reinforce our understanding.

• A. $\frac{4}{9}$: This might arise from incorrectly calculating the exponents or perhaps confusing the bases and exponents. Remember, it's $3^4$ and $2^2$, not the other way around.
• C. $\frac{1}{36}$: This could result from a combination of errors, possibly involving adding the exponents or misinterpreting the negative signs.
• D. $\frac{1}{3}$: This is quite far off and likely indicates a significant misunderstanding of how negative exponents work. Always double-check your exponent rules!

Key Takeaways for Mastering Exponents

To really nail problems like these, here are some key takeaways:

• Understand Negative Exponents Inside and Out: Know what they mean and how to manipulate them. Practice converting expressions with negative exponents to their positive counterparts.
• Master the Order of Operations: Always follow the correct order (PEMDAS/BODMAS) to avoid calculation errors. Exponents come before multiplication and division.
• Practice, Practice, Practice: The more you practice, the more comfortable you'll become with these types of problems. Try different variations and challenge yourself.
• Double-Check Your Work: It's easy to make a small mistake, especially with exponents. Take a moment to review your steps and ensure you haven't made any careless errors.

By following these tips, you'll be well on your way to mastering exponents and tackling even more complex problems with confidence. Remember, math is a journey, not a destination. Enjoy the ride! Guys, keep up the awesome work, and you'll be acing those math problems in no time!

Conclusion

The correct answer is B. $\frac{4}{81}$. We found this by understanding negative exponents, substituting the given values, and simplifying the resulting expression. Whether you prefer to deal with the negative exponents directly or move the terms around first, the key is to apply the rules correctly and carefully. Keep practicing, and you'll become a pro at these types of problems! Remember understanding the underlying concepts is more important than just memorizing steps! Keep rocking it!