Calculating E(a, B) With A * B = 2^12: A Math Challenge

Hey guys! Let's dive into a cool math problem today. We're going to tackle a complex expression and figure out its value. Get ready, because it's going to be a fun ride!

Understanding the Expression E(a, b)

So, the main challenge here is to calculate the value of the expression E(a, b). This expression looks quite intimidating at first glance, but don't worry, we'll break it down step by step. First, let’s write down the expression:

E(a, b) = ( (ab³⁾(1/4) + (a^3 b)^(1/4) ) / (sqrt(a) + sqrt(b)) + (1-sqrt(ab)) / (ab)^(1/4) ) * ( (a b^(1/3) - (ab)^(1/3) ) / ( a^(1/3)-1 ) - ( a b^(1/3) + (a^2 b)^(1/3) ) / ( a^(1/3)+1 ) )

We are also given that a * b = 2^12. This piece of information is crucial and will help us simplify the expression significantly. Our main goal is to simplify this expression and find a numerical value, making use of the given condition. Remember, math problems like these often involve clever algebraic manipulations and recognizing patterns. It's like a puzzle, and we're here to solve it together!

Now, let's start by focusing on the first part of the expression. We'll look at how we can simplify the terms inside the brackets. The key here is to use the properties of exponents and radicals to make the expression more manageable. Sometimes, rewriting the terms in a different form can reveal hidden simplifications. So, stay tuned as we start unraveling this mathematical mystery!

Simplifying the First Part of E(a, b)

The first part of our expression E(a, b) is: ((ab³⁾(1/4) + (a^3 b)^(1/4)) / (sqrt(a) + sqrt(b)) + (1-sqrt(ab)) / (ab)^(1/4). Let's break this down further. We'll start by looking at the numerator of the first term, which is (ab³⁾(1/4) + (a^3 b)^(1/4). We can rewrite these terms using exponent rules. Recall that (xy)^z = x^z * y^z and x^(1/n) is the nth root of x.

So, (ab³⁾(1/4) becomes a^(1/4) * b^(3/4) and (a^3 b)^(1/4) becomes a^(3/4) * b^(1/4). Now, let's factor out the common terms. We can factor out a^(1/4) * b^(1/4) from both terms. This gives us a^(1/4) * b^(1/4) * (b^(1/2) + a^(1/2)). Notice that b^(1/2) is the same as sqrt(b) and a^(1/2) is the same as sqrt(a). So, we now have a^(1/4) * b^(1/4) * (sqrt(b) + sqrt(a)).

The denominator of the first term is (sqrt(a) + sqrt(b)). Now, we can see that the term (sqrt(a) + sqrt(b)) appears in both the numerator and the denominator, so we can cancel them out! This leaves us with a^(1/4) * b^(1/4), which is the same as (ab)^(1/4). How cool is that? We've already simplified a big chunk of the expression. Next, we'll move on to the second term in the first part of E(a, b), which is (1-sqrt(ab)) / (ab)^(1/4). This term looks a bit simpler, but we'll still need to see how it interacts with the rest of the expression. Let's keep pushing forward!

Working on the Second Part of E(a, b)

Now, let’s shift our focus to the second part of the expression E(a, b). This part is: ( (a b^(1/3) - (ab)^(1/3) ) / ( a^(1/3)-1 ) - ( a b^(1/3) + (a^2 b)^(1/3) ) / ( a^(1/3)+1 ) ). This looks a bit more complex, but we'll tackle it with the same strategy: breaking it down and simplifying step by step. First, let's focus on the first fraction within this part: (a b^(1/3) - (ab)^(1/3)) / (a^(1/3) - 1).

In the numerator, a b^(1/3) - (ab)^(1/3), we can rewrite (ab)^(1/3) as a^(1/3) * b^(1/3). So, the numerator becomes a b^(1/3) - a^(1/3) b^(1/3). We can factor out a common term here, which is b^(1/3). Factoring out b^(1/3) gives us b^(1/3) * (a - a^(1/3)). Now, let's look at the second fraction: (a b^(1/3) + (a^2 b)^(1/3)) / (a^(1/3) + 1). Again, we can rewrite (a^2 b)^(1/3) as a^(2/3) * b^(1/3). So, the numerator becomes a b^(1/3) + a^(2/3) b^(1/3). We can factor out a common term here as well, which is b^(1/3). Factoring out b^(1/3) gives us b^(1/3) * (a + a^(2/3)).

Now, we have two simplified fractions. Our next step is to combine these fractions and see if we can simplify further. This might involve finding a common denominator and performing some algebraic manipulations. Remember, the goal is to make this part of the expression as simple as possible so we can eventually multiply it with the first part we simplified earlier. Let’s keep going!

Combining and Simplifying Further

Alright, guys, let's bring those simplified fractions together and see if we can make some more magic happen. We've got: b^(1/3) * (a - a^(1/3)) / (a^(1/3) - 1) - b^(1/3) * (a + a^(2/3)) / (a^(1/3) + 1). To combine these fractions, we need a common denominator. The common denominator here is (a^(1/3) - 1) * (a^(1/3) + 1).

So, let's rewrite each fraction with this common denominator. The first fraction becomes: [b^(1/3) * (a - a^(1/3)) * (a^(1/3) + 1)] / [(a^(1/3) - 1) * (a^(1/3) + 1)]. The second fraction becomes: [b^(1/3) * (a + a^(2/3)) * (a^(1/3) - 1)] / [(a^(1/3) - 1) * (a^(1/3) + 1)]. Now we can combine the numerators: b^(1/3) * [(a - a^(1/3)) * (a^(1/3) + 1) - (a + a^(2/3)) * (a^(1/3) - 1)] / [(a^(1/3) - 1) * (a^(1/3) + 1)]

Let's expand the terms in the numerator. Remember, it's crucial to be careful with the signs here. Expanding and simplifying, we get: b^(1/3) * [a^(4/3) + a - a^(2/3) - a^(1/3) - (a^(4/3) - a - a^(2/3) + a^(1/3))] / [(a^(1/3) - 1) * (a^(1/3) + 1)]

Simplifying further, many terms cancel out, and we're left with: b^(1/3) * [2a - 2a^(1/3)] / [(a^(1/3) - 1) * (a^(1/3) + 1)]

We can factor out 2a^(1/3) from the numerator: 2 * a^(1/3) * b^(1/3) * (a^(2/3) - 1) / (a^(2/3) - 1). Now, we can cancel out the (a^(2/3) - 1) term! This leaves us with 2 * a^(1/3) * b^(1/3). Wow, that's a lot simpler than where we started! Next, we'll combine this simplified expression with the first part of E(a, b) that we simplified earlier.

Putting It All Together

Okay, folks, this is where we bring everything together. We've simplified both parts of E(a, b), and now it's time to multiply them. Remember, the first part simplified to (ab)^(1/4) + (1 - sqrt(ab)) / (ab)^(1/4), and the second part simplified to 2 * a^(1/3) * b^(1/3). Let's multiply these two expressions:

E(a, b) = [ (ab)^(1/4) + (1 - sqrt(ab)) / (ab)^(1/4) ] * [ 2 * a^(1/3) * b^(1/3) ]

First, let's distribute the second term across the terms in the first part: E(a, b) = 2 * (ab)^(1/4) * a^(1/3) * b^(1/3) + 2 * [ (1 - sqrt(ab)) / (ab)^(1/4) ] * a^(1/3) * b^(1/3)

This looks a bit messy, but we can simplify it further by using the properties of exponents. Let's focus on the first term: 2 * (ab)^(1/4) * a^(1/3) * b^(1/3). We can rewrite this as 2 * a^(1/4) * b^(1/4) * a^(1/3) * b^(1/3). Combining the exponents for a and b, we get: 2 * a^(7/12) * b^(7/12) = 2 * (ab)^(7/12)

Now, let's look at the second term: 2 * [ (1 - sqrt(ab)) / (ab)^(1/4) ] * a^(1/3) * b^(1/3). We can rewrite this as: 2 * (1 - sqrt(ab)) * a^(1/3) * b^(1/3) / (a^(1/4) * b^(1/4)). Simplifying further, we get: 2 * (1 - sqrt(ab)) * a^(1/12) * b^(1/12) = 2 * (1 - sqrt(ab)) * (ab)^(1/12)

So, our expression now looks like this: E(a, b) = 2 * (ab)^(7/12) + 2 * (1 - sqrt(ab)) * (ab)^(1/12)

We're getting closer! Now, we need to use the given condition that a * b = 2^12. This is where things will really start to simplify.

Using the Condition a * b = 2^12

Alright, the moment we've been waiting for! Let's use the condition that a * b = 2^12. We'll plug this into our simplified expression for E(a, b): E(a, b) = 2 * (2¹²⁾(7/12) + 2 * (1 - sqrt(2^12)) * (2¹²⁾(1/12)

Let's simplify each term. First, (2¹²⁾(7/12) is equal to 2^(12 * 7/12) = 2^7, which is 128. So, the first term becomes 2 * 128 = 256. Next, let's look at the second term. sqrt(2^12) is equal to 2^6, which is 64. So, (1 - sqrt(2^12)) becomes (1 - 64) = -63. And (2¹²⁾(1/12) is equal to 2^(12 * 1/12) = 2^1, which is 2. Therefore, the second term becomes 2 * (-63) * 2 = -252. Now, we can add the two terms together: E(a, b) = 256 + (-252) = 4

So, after all that work, we've found that E(a, b) = 4! How awesome is that? We took a very complex expression, broke it down into smaller parts, simplified each part, and then used the given condition to find the final value.

Final Answer and Key Takeaways

Drumroll, please! The final answer to our mathematical challenge is:

E(a, b) = 4

That's it, guys! We did it! We took a seemingly impossible expression and simplified it to a single number. This problem was a fantastic exercise in algebraic manipulation, exponent rules, and problem-solving strategies. Here are a few key takeaways from this problem:

1. Break It Down: Complex expressions can be intimidating, but breaking them down into smaller parts makes them much more manageable.
2. Simplify: Look for opportunities to simplify each part of the expression using algebraic rules and exponent properties.
3. Factor: Factoring out common terms can often reveal hidden simplifications.
4. Use Given Conditions: Don't forget to use any given conditions or information. They are usually crucial for finding the final answer.
5. Stay Organized: Keep your work organized and write down each step clearly. This helps prevent errors and makes it easier to follow your logic.

I hope you enjoyed this mathematical journey as much as I did. Remember, math can be challenging, but it's also incredibly rewarding. Keep practicing, keep exploring, and keep having fun with it!