Solving And Simplifying The Math Expression 2⁴.5⁻³.9² Divided By 8.3⁶.125⁻¹

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Hey everyone! Today, we're going to break down a cool math problem that looks a bit intimidating at first, but trust me, it's totally manageable once we understand the tricks. We're talking about simplifying expressions with exponents and fractions – a fundamental skill in mathematics. Let’s jump right in!

Understanding the Problem

The problem we're tackling is: 2⁴.5⁻³.9² / (8.3⁶.125⁻¹). At first glance, it might seem like a jumble of numbers and symbols, but let’s break it down piece by piece.

Exponents and Their Meanings

First off, let's talk exponents. An exponent tells you how many times a number (the base) is multiplied by itself. For example, 2⁴ means 2 multiplied by itself four times: 2 * 2 * 2 * 2. Easy peasy, right?

Now, what about negative exponents? A negative exponent means we're dealing with the reciprocal of the base raised to the positive exponent. So, 5⁻³ is the same as 1 / (5³), which means 1 / (5 * 5 * 5). Got it? Cool!

Breaking Down the Numerator: 2⁴.5⁻³.9²

Let's start with the top part of our fraction, the numerator: 2⁴.5⁻³.9². We'll handle each term separately:

  • 2⁴: This is 2 multiplied by itself four times, which equals 16 (2 * 2 * 2 * 2 = 16).
  • 5⁻³: As we discussed, this is 1 / (5³). So, 5³ is 5 * 5 * 5 = 125. Therefore, 5⁻³ is 1 / 125.
  • 9²: This is 9 multiplied by itself, which equals 81 (9 * 9 = 81).

Now, let’s put it all together: 2⁴.5⁻³.9² = 16 * (1 / 125) * 81. We'll keep it like this for now and simplify later.

Decoding the Denominator: 8.3⁶.125⁻¹

Now, let’s tackle the bottom part of our fraction, the denominator: 8.3⁶.125⁻¹. This looks a bit trickier, but we'll break it down just like before:

  • 8: We can rewrite 8 as 2³. This will be super helpful later when we try to simplify the expression.
  • 3⁶: This is 3 multiplied by itself six times (3 * 3 * 3 * 3 * 3 * 3), which equals 729.
  • 125⁻¹: This is the same as 1 / 125. Notice anything familiar? We saw 125 earlier when we dealt with 5⁻³. This is a hint that we're on the right track to simplifying things!

So, our denominator becomes: 8.3⁶.125⁻¹ = 2³ * 729 * (1 / 125). Let's keep it like this for now.

Simplifying the Expression: Putting It All Together

Okay, guys, now we've broken down both the numerator and the denominator. Let's put them back into our original fraction:

Original expression: 2⁴.5⁻³.9² / (8.3⁶.125⁻¹)

Becomes: (16 * (1 / 125) * 81) / (2³ * 729 * (1 / 125))

Spotting Opportunities to Simplify

Now comes the fun part – simplifying! Look for common factors in the numerator and denominator that we can cancel out. This is where our hard work in breaking down the numbers pays off.

  • Notice that we have (1 / 125) in both the numerator and the denominator. We can cancel these out! This is like dividing both the top and bottom of the fraction by (1 / 125), which doesn't change the value of the fraction.
  • We also know that 16 is 2⁴ and 8 is 2³. So, we can rewrite our expression using these powers of 2.
  • Also, 81 is 9² and 729 is 3⁶, which are powers of 3. This gives us another avenue for simplification.

Rewriting the Expression

Let's rewrite our expression using these observations:

(16 * (1 / 125) * 81) / (2³ * 729 * (1 / 125))

Becomes: (2⁴ * 81) / (2³ * 729)

Cancelling Common Factors

Now we can cancel out common factors:

  • We have 2⁴ in the numerator and 2³ in the denominator. We can cancel out 2³ from both, leaving us with just 2 in the numerator (2⁴ / 2³ = 2¹ = 2).
  • We have 81 in the numerator and 729 in the denominator. Notice that 729 is 81 * 9. So, we can rewrite 729 as 81 * 9 and cancel out the 81s.

The Simplified Expression

After cancelling out the common factors, our expression looks like this:

2 / 9

That’s it! We've simplified the original expression 2⁴.5⁻³.9² / (8.3⁶.125⁻¹) down to 2 / 9. How cool is that?

Steps to Simplify the Problem

Okay, let's recap the steps we took to solve this problem. This is super useful for tackling similar problems in the future. Remember, practice makes perfect!

  1. Break Down the Problem: Identify the different parts of the expression (numerator and denominator) and focus on each one separately.
  2. Understand Exponents: Make sure you know what exponents mean, including negative exponents (reciprocals).
  3. Simplify Each Term: Calculate the values of each term in the numerator and denominator individually.
  4. Rewrite the Expression: Put the simplified terms back into the original expression.
  5. Spot Opportunities to Simplify: Look for common factors in the numerator and denominator that can be cancelled out.
  6. Cancel Common Factors: Cancel out the common factors to simplify the expression.
  7. Final Simplification: Perform any final calculations to get the simplest form of the expression.

Tips for Mastering Exponents and Fractions

  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with exponents and fractions.
  • Know Your Rules: Memorize the rules of exponents (like how to handle negative exponents, multiplying exponents, etc.).
  • Break It Down: Complex problems become easier when you break them down into smaller, manageable steps.
  • Look for Patterns: Spotting patterns and common factors is key to simplifying expressions.
  • Don't Be Afraid to Make Mistakes: Mistakes are part of the learning process. Learn from them, and keep going!

Why This Matters: Real-World Applications

Okay, you might be thinking,