Ascending Order: √8, 2√[5]4, 3^π & More!

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Hey guys! Today, we're diving into a mathematical challenge that involves arranging numbers in ascending order. It might sound intimidating at first, especially with those radicals and exponents, but don't worry, we'll break it down step by step. We'll tackle two sets of numbers: a) √8, 2√[5]4, 2√[3]5 and b) 3^π, √(3√5), 3^√10, 9√[5]729. So, grab your thinking caps, and let's get started!

Part A: √8, 2√[5]4, 2√[3]5

Okay, so our first set of numbers is √8, 2√[5]4, and 2√[3]5. The key here is to make these numbers comparable. Right now, they look pretty different, right? We've got a square root, a fifth root multiplied by 2, and a cube root multiplied by 2. To make things easier, we need to express them with a common exponent or a common root.

Step 1: Convert to Exponential Form

Let's start by converting everything to exponential form. Remember that √x is the same as x^(1/2), √[n]x is the same as x^(1/n), and so on. So, we can rewrite our numbers as:

  • √8 = 8^(1/2)
  • 2√[5]4 = 2 * 4^(1/5)
  • 2√[3]5 = 2 * 5^(1/3)

This is a good start, but we still have that pesky '2' multiplying some of the terms. We need to get rid of that to make a fair comparison.

Step 2: Express Everything with a Common Base (if possible)

We can rewrite 8 as 2^3 and 4 as 2^2. This will help us simplify things further:

  • 8^(1/2) = (23)(1/2) = 2^(3/2)
  • 2 * 4^(1/5) = 2 * (22)(1/5) = 2 * 2^(2/5) = 2^(1 + 2/5) = 2^(7/5)
  • 2 * 5^(1/3) – This one is a bit trickier because 5 can't be easily expressed as a power of 2. We'll come back to this.

Now we have 2^(3/2) and 2^(7/5). These are comparable because they have the same base (2). To compare them, we just need to compare their exponents.

Step 3: Comparing Exponents

Let's compare 3/2 and 7/5. To do this easily, we can find a common denominator. The least common multiple of 2 and 5 is 10, so:

  • 3/2 = 15/10
  • 7/5 = 14/10

Since 15/10 is greater than 14/10, we know that 2^(3/2) is greater than 2^(7/5). This means √8 is greater than 2√[5]4. Awesome!

Step 4: Dealing with 2√[3]5

Now, let's bring back 2√[3]5, which is 2 * 5^(1/3). This is where things get a little less straightforward. We can't directly compare it to the powers of 2 we have. One way to compare is to try and raise everything to a power that eliminates the fractional exponents. We have exponents 1/2, 1/5, and 1/3. The least common multiple of 2, 5, and 3 is 30. So, let's raise all the original numbers to the power of 30:

  • (√8)^30 = (8(1/2))30 = 8^15 = (23)15 = 2^45
  • (2√[5]4)^30 = (2 * 4(1/5))30 = 2^30 * (22)(30/5) = 2^30 * 2^12 = 2^42
  • (2√[3]5)^30 = (2 * 5(1/3))30 = 2^30 * 5^(30/3) = 2^30 * 5^10

Now we need to compare 2^45, 2^42, and 2^30 * 5^10. This still looks a bit tricky, but we've made progress. 2^45 is clearly the largest so far. We need to figure out how 2^30 * 5^10 relates to 2^42.

Step 5: Final Comparison

Let's rewrite 2^42 as 2^30 * 2^12. Now we are comparing 2^30 * 5^10 with 2^30 * 2^12. We can divide both by 2^30 and we end up comparing 5^10 and 2^12.

  • 5^10 = (55)2 = 3125^2
  • 2^12 = (26)2 = 64^2

Clearly, 3125^2 is much larger than 64^2. So, 2^30 * 5^10 is greater than 2^42.

Step 6: Ascending Order

Now we know:

  • (2√[5]4)^30 < (2√[3]5)^30 < (√8)^30

Since all the numbers are positive, raising them to the power of 30 doesn't change their order. Therefore, the ascending order is:

2√[5]4 < 2√[3]5 < √8

Phew! That was a workout, guys. But we got there in the end! Now, let's tackle the second set of numbers.

Part B: 3^π, √(3√5), 3^√10, 9√[5]729

Alright, time to tackle the next set of numbers: 3^π, √(3√5), 3^√10, and 9√[5]729. This looks like another fun challenge! We've got exponents, radicals, and a mix of bases and powers. Just like before, our goal is to make these numbers comparable.

Step 1: Simplify the Radicals

Let's start by simplifying the radicals as much as possible. We have √(3√5) and 9√[5]729. Let's break them down:

  • √(3√5) = (3 * (35)(1/5))^(1/2) = (3 * 3)^(1/2) = (32)(1/2) = 3^(2/2) = 3
  • 9√[5]729 = 9 * (729)^(1/5) = 3^2 * (36)(1/5) = 3^2 * 3^(6/5) = 3^(2 + 6/5) = 3^(16/5)

Wow, that simplifies things quite a bit! Now our numbers are:

  • 3^π
  • 3
  • 3^√10
  • 3^(16/5)

Step 2: Comparing Exponents

Now we have a common base (3) for all the numbers, which is excellent! To arrange them in ascending order, we just need to compare their exponents:

  • π (approximately 3.14159)
  • 1 (since 3 = 3^1)
  • √10 (approximately 3.16228)
  • 16/5 = 3.2

Step 3: Ordering the Exponents

Now we just need to put these exponents in ascending order:

1 < π < √10 < 16/5

Step 4: Ascending Order of the Numbers

Since the base is the same (3), the order of the exponents directly corresponds to the order of the numbers. Therefore, the ascending order is:

3 < 3^π < 3^√10 < 3^(16/5)

Or, in the original form:

√(3√5) < 3^π < 3^√10 < 9√[5]729

And there you have it, guys! We've successfully arranged both sets of numbers in ascending order. The key is to simplify, find common bases or exponents, and then compare carefully. These kinds of problems might seem tough at first, but with a little practice, you'll be a pro in no time!

Key Takeaways

  • Convert to exponential form: Radicals can be tricky to compare, so convert them to exponential form (x^(1/n)) to make things easier.
  • Find a common base: If possible, express all numbers with the same base. This allows you to compare the exponents directly.
  • Simplify: Simplify radicals and exponents as much as possible before comparing.
  • Raise to a common power: If you have different fractional exponents, raising all numbers to a common power (the least common multiple of the denominators) can help eliminate the fractions and make comparison easier.
  • Approximate when needed: For numbers like π and √10, it's often helpful to use approximations to make comparisons.

Practice Makes Perfect

Remember, guys, math is like a muscle – the more you exercise it, the stronger it gets! Try working through similar problems, and don't be afraid to ask for help if you get stuck. Keep practicing, and you'll become a master at arranging numbers in ascending order!

Hope this helped you guys out! Let me know if you have any other math questions you'd like me to tackle. Keep those brains buzzing!