Ascending Order: Sorting Sets T, C, And B

Hey guys! Today, we're diving into a fun math problem where we need to sort elements within different sets in ascending order. This means we'll be arranging the numbers from smallest to largest. We've got three sets to tackle: T, C, and B. Each set contains numbers expressed with square roots, which might seem a bit tricky at first, but don't worry, we'll break it down step by step.

Understanding the Sets: T, C, and B

Before we jump into the sorting, let's take a closer look at the sets we're working with. Understanding the structure of each set is crucial for accurately ordering the elements. The sets are defined as follows:

• Set T: {4√5, 5√2, 2√7, 2√2, 6√2, 4√3}
• Set C: {4√3, 3√5, 6√7, 5√6, 8√2, 5√7}
• Set B: {9√2, 7√6, 3√6, 2√11, 8√5, 4√7}

Notice that each element in the sets is a number multiplied by the square root of another number. To compare these elements, we'll need to find a way to express them in a comparable form. The key here is to bring the numbers outside the square root inside by squaring them. For example, to compare 4√5, we can rewrite it as √(4² * 5) = √(16 * 5) = √80. This way, we can directly compare the values under the square root.

Converting to Comparable Form

Let's convert all the elements in each set to this comparable form. This will make it much easier to see the actual values and arrange them in ascending order. This process involves squaring the coefficient (the number outside the square root) and multiplying it by the number inside the square root.

For example:

• a√b becomes √(a² * b)

We will apply this to each element in sets T, C, and B. This transformation allows us to compare the numbers directly by looking at the values inside the square roots. This is because the square root function is monotonically increasing, meaning that if x < y, then √x < √y. By converting all numbers into this format, we ensure a clear and accurate comparison.

Why This Method Works

Converting the numbers into the form √(a² * b) simplifies the comparison because it eliminates the coefficient outside the square root. By doing this, we transform each number into a form where the entire value is represented under the square root. The magnitude of the number under the square root then directly corresponds to the magnitude of the original number. This is a crucial step because it provides a visual and numerical basis for ordering the numbers. Without this conversion, comparing numbers like 4√5 and 5√2 directly would be difficult because we would have to estimate the square roots and perform multiplications, which can be prone to errors. This method streamlines the process and minimizes the chances of making mistakes.

Sorting Set T

Okay, let's start sorting! We'll begin with Set T. First, we need to convert each element into its comparable form (√a). This will make it much easier to see which numbers are smaller and which are larger. Remember, we do this by squaring the number outside the square root and multiplying it by the number inside.

Set T: {4√5, 5√2, 2√7, 2√2, 6√2, 4√3}

1. 4√5 = √(4² * 5) = √(16 * 5) = √80
2. 5√2 = √(5² * 2) = √(25 * 2) = √50
3. 2√7 = √(2² * 7) = √(4 * 7) = √28
4. 2√2 = √(2² * 2) = √(4 * 2) = √8
5. 6√2 = √(6² * 2) = √(36 * 2) = √72
6. 4√3 = √(4² * 3) = √(16 * 3) = √48

Now we have Set T represented as: {√80, √50, √28, √8, √72, √48}.

Arranging Set T in Ascending Order

Now that we've converted the elements into a comparable format, we can easily arrange them in ascending order. We simply look at the numbers under the square root and order them from smallest to largest.

Looking at the values: √80, √50, √28, √8, √72, √48, we can arrange them as follows:

√8 < √28 < √48 < √50 < √72 < √80

Converting these back to their original form, we get the sorted set T:

{2√2, 2√7, 4√3, 5√2, 6√2, 4√5}

So, that's it! We've successfully sorted Set T in ascending order. The process of converting to comparable form made it much easier to identify the correct order. Now, let's move on to Set C.

Sorting Set C

Next up is Set C. We'll follow the same process we used for Set T: convert each element to its comparable form and then arrange them in ascending order. This consistent approach helps ensure accuracy and efficiency.

Set C: {4√3, 3√5, 6√7, 5√6, 8√2, 5√7}

Let's convert each element:

1. 4√3 = √(4² * 3) = √(16 * 3) = √48
2. 3√5 = √(3² * 5) = √(9 * 5) = √45
3. 6√7 = √(6² * 7) = √(36 * 7) = √252
4. 5√6 = √(5² * 6) = √(25 * 6) = √150
5. 8√2 = √(8² * 2) = √(64 * 2) = √128
6. 5√7 = √(5² * 7) = √(25 * 7) = √175

Now Set C is represented as: {√48, √45, √252, √150, √128, √175}.

Arranging Set C in Ascending Order

Time to arrange these values in ascending order. Looking at the numbers under the square root, it should be pretty straightforward.

Values: √48, √45, √252, √150, √128, √175. Arranged in ascending order:

√45 < √48 < √128 < √150 < √175 < √252

Converting back to the original form, we get the sorted Set C:

{3√5, 4√3, 8√2, 5√6, 5√7, 6√7}

Awesome! Set C is sorted. We're on a roll! Just one more set to go – Set B. Let's keep the momentum going and tackle the final set with the same systematic approach.

Sorting Set B

Alright, last but not least, we have Set B to sort. By now, you're probably getting the hang of this. We'll follow the same tried-and-true method: convert to comparable form, then arrange in ascending order.

Set B: {9√2, 7√6, 3√6, 2√11, 8√5, 4√7}

Let's convert those elements:

1. 9√2 = √(9² * 2) = √(81 * 2) = √162
2. 7√6 = √(7² * 6) = √(49 * 6) = √294
3. 3√6 = √(3² * 6) = √(9 * 6) = √54
4. 2√11 = √(2² * 11) = √(4 * 11) = √44
5. 8√5 = √(8² * 5) = √(64 * 5) = √320
6. 4√7 = √(4² * 7) = √(16 * 7) = √112

So, Set B is now represented as: {√162, √294, √54, √44, √320, √112}.

Arranging Set B in Ascending Order

Time for the final sorting! Let's arrange the numbers under the square roots in ascending order.

Values: √162, √294, √54, √44, √320, √112. In ascending order:

√44 < √54 < √112 < √162 < √294 < √320

Converting back to the original form, we get the sorted Set B:

{2√11, 3√6, 4√7, 9√2, 7√6, 8√5}

Final Sorted Sets

We did it! We've successfully sorted all three sets in ascending order. Let's take a look at the final results:

• Set T: {2√2, 2√7, 4√3, 5√2, 6√2, 4√5}
• Set C: {3√5, 4√3, 8√2, 5√6, 5√7, 6√7}
• Set B: {2√11, 3√6, 4√7, 9√2, 7√6, 8√5}

Recap of the Process

To recap, the key to sorting these sets was to convert each element into a comparable form by expressing them as √(a² * b). This allowed us to directly compare the values under the square root and easily arrange them in ascending order. This method is super helpful when you're dealing with numbers that aren't immediately comparable.

Conclusion

Sorting sets with square roots can seem intimidating at first, but as we've seen, a systematic approach makes it totally manageable. By converting the elements to a comparable form, we simplified the process and ensured accurate results. This method isn't just useful for math problems; it's a great way to develop problem-solving skills that can be applied in many areas of life. Keep practicing, and you'll become a pro at sorting all kinds of numbers! Great job, guys!