Ascending Order $\sqrt[4]{3}, \sqrt[6]{7}, \sqrt[12]{48}$

Hey there, math enthusiasts! Today, we're diving into a fascinating problem: arranging radicals in ascending order. Specifically, we'll be tackling the challenge of comparing $\sqrt[4]{3}, \sqrt[6]{7}$, and $\sqrt[12]{48}$. This might seem tricky at first glance, but don't worry, we'll break it down step-by-step. Radicals, those mathematical expressions involving roots, can sometimes appear daunting when you're asked to compare them. The key to unraveling these comparisons lies in finding a common ground. In the realm of radicals, that common ground is often the index of the root. To truly grasp the concept of ordering radicals, it's essential to have a solid understanding of what radicals represent. Remember, a radical expression like $\sqrt[n]{a}$ is asking, "What number, when raised to the power of n, gives you a?" The index n tells us the type of root we're dealing with – a square root (n=2), a cube root (n=3), a fourth root (n=4), and so on. The radicand, a, is the number or expression under the radical sign. When radicals have different indices, it's like comparing apples and oranges. We need a way to express them in a uniform manner before we can make a meaningful comparison. This is where the concept of finding a common index comes into play. The strategy we'll employ involves transforming the radicals into equivalent forms that share the same index. Once we have a common index, we can directly compare the radicands – the numbers under the radical sign – to determine the order. Think of it like converting fractions to a common denominator before comparing their sizes.

Finding the Least Common Multiple (LCM) of Indices

The first crucial step in arranging radicals in ascending order is identifying the indices. In our case, we have indices of 4, 6, and 12. To compare these radicals effectively, we need to find the least common multiple (LCM) of these indices. The LCM is the smallest number that is a multiple of all the given numbers. Finding the LCM is like finding the smallest common denominator when you're adding fractions. It's the key to bringing our radicals to a level playing field. The least common multiple (LCM) of the indices will serve as our new common index. This will allow us to rewrite each radical with the same root, making comparisons much simpler. So, how do we find the LCM of 4, 6, and 12? There are a couple of methods we can use. One common approach is to list out the multiples of each number until we find a common one. Another method, often more efficient for larger numbers, is to use prime factorization. Let's break down each number into its prime factors:

• 4 = 2 × 2 = $2^2$
• 6 = 2 × 3
• 12 = 2 × 2 × 3 = $2^2$ × 3

To find the LCM, we take the highest power of each prime factor that appears in any of the factorizations. In this case, the highest power of 2 is $2^2$, and the highest power of 3 is 3. So, the LCM is $2^2$ × 3 = 4 × 3 = 12. This means that 12 is the smallest number that is divisible by 4, 6, and 12. This LCM, 12, will be our common index. Now, we can move on to rewriting each radical with this new index. By finding the LCM, we've essentially discovered the magic number that will allow us to transform our radicals into comparable forms. This is a fundamental step in solving problems involving radical comparisons, and it's a skill that will come in handy in various mathematical contexts.

Converting Radicals to a Common Index

Now that we've found the LCM of the indices (which is 12), our next task is to convert each radical to an equivalent form with an index of 12. This conversion involves adjusting both the index and the radicand (the number inside the radical) in a specific way to maintain the value of the expression. Think of it like scaling a recipe – you need to adjust all the ingredients proportionally to keep the taste the same. To convert a radical, we multiply the index by a certain factor to get our common index (12). We then raise the radicand to the power of that same factor. This ensures that we're not changing the overall value of the radical, just its representation. Let's apply this process to each of our radicals:

1. $\sqrt[4]{3}$ We need to multiply the index 4 by 3 to get 12 (4 × 3 = 12). So, we raise the radicand 3 to the power of 3 as well:

   $$\sqrt[4]{3} = \sqrt[4 \times 3]{3^3} = \sqrt[12]{27}$$

2. $\sqrt[6]{7}$ We need to multiply the index 6 by 2 to get 12 (6 × 2 = 12). So, we raise the radicand 7 to the power of 2:

   $$\sqrt[6]{7} = \sqrt[6 \times 2]{7^2} = \sqrt[12]{49}$$

3. $\sqrt[12]{48}$ This radical already has an index of 12, so we don't need to change it:

   $$\sqrt[12]{48}$$

By performing these conversions, we've successfully transformed our original radicals into equivalent expressions with a common index of 12. This is a crucial step because it allows us to directly compare the radicands and determine the order of the radicals. Now that we have $\sqrt[12]{27}$, $\sqrt[12]{49}$, and $\sqrt[12]{48}$, the comparison becomes much clearer.

Comparing Radicands and Ordering

With all the radicals now sharing a common index of 12, the final step in arranging radicals in ascending order is a breeze! We can now directly compare the radicands – the numbers under the radical sign. Remember, when radicals have the same index, the radical with the smaller radicand is the smaller number. It's like comparing fractions with the same denominator – the fraction with the smaller numerator is the smaller fraction. Looking at our converted radicals, we have:

• $\sqrt[12]{27}$
• $\sqrt[12]{49}$
• $\sqrt[12]{48}$

The radicands are 27, 49, and 48. Arranging these in ascending order is straightforward: 27 < 48 < 49. Therefore, we can conclude that: $\sqrt[12]{27} < \sqrt[12]{48} < \sqrt[12]{49}$ But we're not quite done yet! We need to express our answer in terms of the original radicals. So, let's substitute back the original forms: $\sqrt[4]{3} < \sqrt[12]{48} < \sqrt[6]{7}$ And there you have it! We've successfully arranged the radicals in ascending order. This process highlights the power of transforming expressions to a common base for easy comparison. By finding the LCM of the indices and converting the radicals, we were able to reduce the problem to a simple comparison of numbers. This approach is applicable to a wide range of radical comparison problems, making it a valuable tool in your mathematical arsenal. Remember, the key is to find that common ground – in this case, the common index – to make the comparison clear and straightforward.

Final Answer: Ascending Order of Radicals

So, let's recap what we've done, guys! We started with the seemingly tricky task of arranging the radicals $\sqrt[4]{3}, \sqrt[6]{7}$, and $\sqrt[12]{48}$ in ascending order. To tackle this, we followed a systematic approach:

1. Found the LCM of the indices: The indices were 4, 6, and 12, and their LCM is 12. This became our common index.
2. Converted radicals to the common index: We transformed each radical into an equivalent form with an index of 12. This involved raising the radicand to the appropriate power.
3. Compared radicands: Once all radicals had the same index, we simply compared the numbers under the radical sign (the radicands).
4. Ordered the radicals: Based on the comparison of radicands, we arranged the radicals in ascending order.
5. Expressed the answer in original form: Finally, we substituted back the original radical expressions to present our final answer.

Following these steps, we confidently determined that the ascending order of the given radicals is: $\sqrt[4]{3} < \sqrt[12]{48} < \sqrt[6]{7}$ This exercise demonstrates a powerful technique for comparing and ordering radicals. By finding a common index, we can transform complex expressions into easily comparable forms. This skill is not only useful for solving mathematical problems but also for developing a deeper understanding of radical expressions and their properties. Remember, the key to success in math often lies in breaking down complex problems into smaller, manageable steps. By systematically applying the principles of LCM and radical conversion, we were able to conquer this challenge. Keep practicing, and you'll become a radical ordering pro in no time! This method not only helps in solving mathematical problems but also enhances our understanding of number properties and comparisons, which is a fundamental aspect of mathematics.