Solving Exponential Equation: Step-by-Step Guide

Hey guys! Let's dive into solving an interesting exponential equation today. We're going to break down the steps to tackle this problem: xx = 8^33 / [4^33 * 2^34 + (2^5 * (2²⁰⁾5) / (16 * 2^23) + 7^5 / (7^5 * 1^32 * 4)]. This might look intimidating at first, but don't worry, we'll go through it together, making sure every step is clear. This article is designed to make complex math problems understandable, even if you're just starting out with exponents and powers. So, grab your calculators (or your mental math muscles!), and let’s get started!

Breaking Down the Equation

First off, let's rewrite the equation to make it super clear what we're working with:

xx = 8^33 / [4^33 * 2^34 + (2^5 * (2²⁰⁾5) / (16 * 2^23) + 7^5 / (7^5 * 1^32 * 4)]

Our main goal here is to simplify this massive expression piece by piece. We'll start by tackling the exponents and then move on to simplifying the larger terms within the brackets. Remember, the key to solving complex equations is to break them down into smaller, more manageable parts. This way, you avoid getting overwhelmed and can focus on each operation individually. We'll use the rules of exponents extensively, such as a^(m+n) = a^m * a^n and (a^m)n = a^(m*n), so having a good grasp of these rules is essential. Don't worry if you need to brush up on them – we'll explain each step as we go along!

Step 1: Simplify the Terms Inside the Brackets

Let's focus on the terms within the square brackets. This is where most of the action is happening! We have a few different operations going on here: multiplication, division, and addition. Remember our order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. We'll follow this order to make sure we simplify correctly.

Term 1: 4^33 * 2^34

To simplify this, we need to express both terms with the same base. Since 4 is 2 squared (4 = 2^2), we can rewrite 4^33 as (2²⁾33. Using the power of a power rule, this becomes 2^(2*33) = 2^66. Now we have:

2^66 * 2^34

Using the rule for multiplying exponents with the same base (a^m * a^n = a^(m+n)), we add the exponents:

2^(66+34) = 2^100

So, the first term simplifies to 2^100. Not too shabby, right? We’re making progress already!

Term 2: (2^5 * (2²⁰⁾5) / (16 * 2^23)

This term looks a bit more complex, but we'll break it down step by step. First, let's deal with the power of a power: (2²⁰⁾5. Again, using the rule (a^m)n = a^(m*n), we get:

(2²⁰⁾5 = 2^(20*5) = 2^100

Now we have:

(2^5 * 2^100) / (16 * 2^23)

Next, we simplify the numerator by multiplying the exponents with the same base:

2^5 * 2^100 = 2^(5+100) = 2^105

Now our term looks like this:

2^105 / (16 * 2^23)

We need to express 16 as a power of 2. Since 16 = 2^4, we can rewrite the term as:

2^105 / (2^4 * 2^23)

Simplify the denominator:

2^4 * 2^23 = 2^(4+23) = 2^27

Now we have:

2^105 / 2^27

Using the rule for dividing exponents with the same base (a^m / a^n = a^(m-n)), we subtract the exponents:

2^(105-27) = 2^78

So, the second term simplifies to 2^78. We’re on a roll!

Term 3: 7^5 / (7^5 * 1^32 * 4)

This term looks simpler compared to the others. We can immediately see that 7^5 in the numerator and denominator will cancel each other out:

7^5 / 7^5 = 1

Also, 1 raised to any power is just 1, so 1^32 = 1. Now we have:

1 / (1 * 4) = 1 / 4

We can also express 1/4 as a power of 2. Since 4 = 2^2, then 1/4 = 2^(-2). So, the third term simplifies to 2^(-2).

Step 2: Combine the Simplified Terms

Now that we've simplified each term inside the brackets, let's put them back together. Remember, our original expression inside the brackets was:

[4^33 * 2^34 + (2^5 * (2²⁰⁾5) / (16 * 2^23) + 7^5 / (7^5 * 1^32 * 4)]

We've simplified each part, so now we have:

[2^100 + 2^78 + 2^(-2)]

To add these terms, it would be ideal to have a common factor. Notice that 2^(-2) is significantly smaller than 2^100 and 2^78. In many practical scenarios, we might consider 2^(-2) negligible in comparison. However, for the sake of precision, let's keep it for now.

To proceed, we can factor out the smallest power of 2, which is 2^(-2):

2^(-2) * [2^(100 - (-2)) + 2^(78 - (-2)) + 1]

This simplifies to:

2^(-2) * [2^102 + 2^80 + 1]

Now we have the expression inside the brackets simplified. Let's move on to the next step.

Step 3: Substitute Back into the Original Equation

Our original equation was:

xx = 8^33 / [4^33 * 2^34 + (2^5 * (2²⁰⁾5) / (16 * 2^23) + 7^5 / (7^5 * 1^32 * 4)]

We've simplified the term inside the brackets to:

2^(-2) * [2^102 + 2^80 + 1]

So, substituting this back into the equation, we get:

xx = 8^33 / (2^(-2) * [2^102 + 2^80 + 1])

Now, we need to express 8 as a power of 2. Since 8 = 2^3, we can rewrite 8^33 as (2³⁾33:

(2³⁾33 = 2^(3*33) = 2^99

So, our equation now looks like this:

xx = 2^99 / (2^(-2) * [2^102 + 2^80 + 1])

Step 4: Simplify the Division

To simplify the division, we can rewrite the expression as:

xx = 2^99 * (1 / (2^(-2) * [2^102 + 2^80 + 1]))

Which is the same as:

xx = 2^99 * (2^2 / [2^102 + 2^80 + 1])

This simplifies to:

xx = (2^(99+2)) / [2^102 + 2^80 + 1]

xx = 2^101 / [2^102 + 2^80 + 1]

Step 5: Approximate the Result

At this point, we have a simplified expression, but it's still quite large. To get an approximate value for xx, we can think about the magnitudes of the numbers involved. The term 2^102 is significantly larger than 2^80 and 1, so we can approximate the denominator as 2^102. This gives us:

xx ≈ 2^101 / 2^102

Using the rule for dividing exponents with the same base, we get:

xx ≈ 2^(101-102) = 2^(-1)

xx ≈ 1/2

So, the approximate value of xx is 1/2 or 0.5.

Conclusion

Wow, we made it through! We started with a complex exponential equation and, by breaking it down step by step, we were able to simplify it and find an approximate solution. Remember, the key to solving these types of problems is to:

1. Break the problem into smaller parts.
2. Apply the rules of exponents.
3. Simplify each part systematically.
4. Combine the simplified parts.
5. Approximate when necessary.

I hope this guide has been helpful and has made solving exponential equations a little less daunting. Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to try another problem, just let me know. Happy solving!