Math Expressions: Match The Answer

Hey guys! Today, we're diving deep into the awesome world of math to tackle some cool expressions. If you're looking to sharpen your skills or just want a fun brain workout, you've come to the right place. We've got some expressions involving roots and exponents, and our mission, should we choose to accept it, is to match each one with its correct value. It’s like a puzzle, but with numbers!

We'll be working with three main expressions:

• I. ⁶√7⁶
• II. ¹²√6²⁴
• III. ³√2¹²

And we have four potential answers to choose from:

• A. 36
• B. 16
• C. 7
• D. 6

Our goal is to figure out which answer corresponds to which expression. Ready to get started? Let's break down each expression one by one. This isn't just about getting the right answer; it's about understanding why it's the right answer. We'll explore the properties of exponents and roots that make these calculations straightforward. So, grab your thinking caps, and let's get this math party started!

Understanding the Basics: Roots and Exponents

Before we jump into solving these specific problems, let's quickly refresh some fundamental concepts about roots and exponents, because knowing these rules is key to unlocking the answers. Think of exponents as a shorthand for repeated multiplication. For example, 7⁶ means 7 multiplied by itself six times (7 * 7 * 7 * 7 * 7 * 7). Roots, on the other hand, are the inverse operation of exponentiation. The nth root of a number 'x' is a value 'y' such that when 'y' is raised to the power of 'n', it equals 'x'.

Mathematically, we often use the property that the nth root of x to the power of n, written as ⁿ√(xⁿ), simplifies to the absolute value of x, |x|, if n is even, and simply x if n is odd. This is because raising a negative number to an even power results in a positive number, so the root needs to account for both positive and negative possibilities. However, in many introductory contexts, especially when dealing with positive bases like we have here, we can simplify this to just x. Another crucial property we'll use is the rule for simplifying roots of powers: ⁿ√(xᵐ) can be rewritten as x^(m/n). This fractional exponent form is super handy for simplifying complex expressions. Remember, these aren't just arbitrary rules; they stem from the very nature of how numbers and operations interact. By mastering these foundational rules, you'll find that solving a wide range of mathematical problems becomes significantly easier and, dare I say, even fun!

Let's take Expression I: ⁶√7⁶. Here, we have the sixth root of 7 raised to the sixth power. According to the property ⁿ√(xⁿ) = x (for positive x), the 6 in the exponent and the 6 in the root effectively cancel each other out. This leaves us with just the base, which is 7. So, ⁶√7⁶ = 7. This means our first expression, I, matches with answer C. 7. Easy peasy, right? This highlights the power of understanding these fundamental properties. When the index of the root matches the exponent of the base inside the radical, the base itself is revealed, assuming it's a positive number.

Now, let's move on to Expression II: ¹²√6²⁴. This one looks a bit more intimidating with larger numbers, but the same rules apply! We can use the property ⁿ√(xᵐ) = x^(m/n). In this case, n = 12, x = 6, and m = 24. So, we can rewrite this as 6^(24/12). When we divide 24 by 12, we get 2. So, the expression simplifies to 6². And what is 6²? It's 6 multiplied by itself, which is 36. Therefore, ¹²√6²⁴ = 36. This means Expression II matches with answer A. 36. See? With the right approach, even complex-looking problems become manageable. It's all about breaking them down using the properties you already know!

Finally, let's tackle Expression III: ³√2¹². Again, we'll use the same rule: ⁿ√(xᵐ) = x^(m/n). Here, n = 3, x = 2, and m = 12. So, we rewrite this as 2^(12/3). Dividing 12 by 3 gives us 4. The expression simplifies to 2⁴. What is 2⁴? It's 2 multiplied by itself four times: 2 * 2 * 2 * 2. That equals 16. So, ³√2¹² = 16. This means Expression III matches with answer B. 16.

So, to recap our findings:

• Expression I (⁶√7⁶) equals 7 (Answer C).
• Expression II (¹²√6²⁴) equals 36 (Answer A).
• Expression III (³√2¹²) equals 16 (Answer B).

This gives us the matching combination: I-C, II-A, III-B. It's always satisfying when all the pieces fall into place!

Solving Expression I: ⁶√7⁶

Alright guys, let's zoom in on the first expression: I. ⁶√7⁶. This problem is a fantastic introduction to how roots and exponents play together. The notation ⁶√ signifies the sixth root, and 7⁶ means 7 raised to the power of 6. When you see a root and an exponent with the same number, especially when the exponent is inside the radical and matches the index of the root, it's usually a sign that things are going to simplify beautifully. The fundamental property at play here is that the nth root and the nth power are inverse operations. Think of it like this: if you square a number and then take the square root, you get back to your original number (assuming it was non-negative). The same logic applies here with the sixth root and the sixth power.

So, in the expression ⁶√7⁶, the sixth root operation essentially 'undoes' the sixth power operation. This leaves us with just the base number, which is 7. Mathematically, we can express this using fractional exponents as well. The sixth root of 7⁶ is equivalent to (7⁶)^(1/6). When you raise a power to another power, you multiply the exponents. So, 6 * (1/6) = 1. This means we have 7¹, which is simply 7. This confirms our result: ⁶√7⁶ = 7. This matches directly with option C. 7. It's a clean and direct application of the definition of roots. It's important to remember that this simplification to just the base works perfectly when the base is positive, as 7 is. If the base were negative and the root index even, we'd have to consider the absolute value, but that's not the case here, making this a straightforward win!

This concept is super important for building confidence in dealing with more complex algebraic manipulations later on. Always look for these inverse relationships between operations; they are the secret shortcuts in math. So, for our first match, we confidently say I corresponds to C. Keep that in mind as we move on to the next challenges!

Solving Expression II: ¹²√6²⁴

Now, let's tackle Expression II: ¹²√6²⁴. This expression involves a higher root index and a larger exponent, but guess what? The principle is exactly the same, and the fractional exponent rule is our best friend here. The rule states that ⁿ√(xᵐ) is equal to x^(m/n). In our expression, the root index 'n' is 12, the base 'x' is 6, and the exponent 'm' is 24.

Applying the rule, we can rewrite ¹²√6²⁴ as 6^(24/12). The next step is to simplify the exponent: 24 divided by 12 equals 2. So, our expression transforms into 6². Now, we just need to calculate 6²: that's 6 multiplied by itself, which is 6 * 6 = 36. Therefore, ¹²√6²⁴ = 36. This makes our second expression match with answer A. 36. Pretty neat, huh? It shows that even with bigger numbers, the underlying mathematical logic remains consistent. You just need to apply the rules correctly.

This type of simplification is incredibly useful. It allows us to take something that looks complicated and reduce it to a much simpler form. For instance, imagine you had to estimate ¹²√6²⁴. Without simplifying, it's tough! But knowing it equals 36 makes it trivial. This skill is essential not just for solving problems but also for understanding the magnitude and behavior of mathematical functions. It's like having a decoder ring for numbers. So, we've established that II corresponds to A. Two down, one to go!

Solving Expression III: ³√2¹²

Last but not least, let's conquer Expression III: ³√2¹². This one involves a cube root (the third root) and a base of 2 raised to the power of 12. Once again, we'll lean on our trusty rule for simplifying radicals with exponents: ⁿ√(xᵐ) = x^(m/n). Here, our root index 'n' is 3, the base 'x' is 2, and the exponent 'm' is 12.

So, we can rewrite ³√2¹² as 2^(12/3). Simplifying the exponent, 12 divided by 3 equals 4. This leaves us with 2⁴. Now, we calculate 2⁴. This means 2 multiplied by itself four times: 2 * 2 * 2 * 2. Let's do the math: 2 * 2 = 4, then 4 * 2 = 8, and finally 8 * 2 = 16. So, ³√2¹² = 16. This means our third expression matches with answer B. 16. Fantastic!

This result highlights how a small base raised to a moderate power can yield a number that might seem surprising if you didn't know the exponent rules. The power of exponents is truly remarkable! It's another example of how the fractional exponent rule simplifies radical expressions. Understanding this rule makes navigating expressions with mixed radicals and powers much more intuitive. It's like having a magic wand that turns complex math into simple arithmetic. Thus, we conclude that III corresponds to B. We've now solved all three expressions and found their corresponding answers. What a journey!

The Final Match: Putting It All Together

So, guys, we've done the heavy lifting and figured out the value of each expression. Let's bring it all together to find the correct matching combination.

• For Expression I (⁶√7⁶), we found the value to be 7. Looking at our options, 7 corresponds to C. So, I-C.
• For Expression II (¹²√6²⁴), we calculated the value to be 36. Among our options, 36 is A. So, II-A.
• For Expression III (³√2¹²), we determined the value to be 16. The option that matches 16 is B. So, III-B.

Putting these together, the correct matching is I-C, II-A, III-B. This is one of the provided combinations. It's always a great feeling when you can systematically solve a problem and arrive at a definitive answer. These kinds of exercises are fundamental building blocks in mathematics, reinforcing the understanding of how different mathematical concepts interact and simplify. Keep practicing these techniques, and you'll find that more advanced topics become much more accessible. Math is all about building a solid foundation, and skills like these are a crucial part of that foundation. Great job working through this with me!