Simplifying The Expression: 128 * (2^-2) * (3^5) * (36^-4)

by Admin 59 views
Simplifying the Expression: 128 * (2^-2) * (3^5) * (36^-4)

Hey guys! Today, we're going to dive deep into simplifying a seemingly complex mathematical expression. It might look intimidating at first glance, but don't worry, we'll break it down step by step and make it super easy to understand. Our mission is to simplify: 128 * (2^-2) * (3^5) * (36^-4). So, grab your calculators (or your mental math hats) and let's get started!

Understanding the Basics

Before we jump into the calculation, let's quickly refresh some key concepts. Remember, when we see exponents, they tell us how many times to multiply a number by itself. For example, 3^5 (3 to the power of 5) means 3 * 3 * 3 * 3 * 3. Also, a negative exponent, like 2^-2, means we take the reciprocal of the base raised to the positive exponent. So, 2^-2 is the same as 1 / (2^2). Understanding these basics is crucial for simplifying any exponential expression.

Another key concept here is breaking down numbers into their prime factors. This is a super handy trick when dealing with exponents, especially when we have expressions like 36^-4. We can express 36 as 6 * 6, which can further be broken down into 2 * 3 * 2 * 3, or 2^2 * 3^2. This allows us to manipulate the expression more easily. So, always remember to look for ways to break down larger numbers into their prime factors – it's a lifesaver!

Step 1: Breaking Down the Expression

Okay, let's start by breaking down our expression piece by piece. We have:

  • 128: We can express 128 as a power of 2. 128 is 2 * 2 * 2 * 2 * 2 * 2 * 2, which is 2^7.
  • (2^-2): This is a negative exponent, so we rewrite it as 1 / (2^2).
  • (3^5): This is already in a simple form, so we'll leave it as is.
  • (36^-4): Remember our trick? Let's break down 36. 36 is 2^2 * 3^2. So, 36^-4 becomes (2^2 * 32)-4.

Now, let's rewrite the entire expression with these breakdowns:

2^7 * (1 / (2^2)) * 3^5 * (2^2 * 32)-4

See? It's already looking a bit more manageable. We've transformed the original expression into something we can work with more easily. The key here is to take it one step at a time, breaking down each component before putting it all back together.

Step 2: Applying the Power of a Product Rule

Next up, we need to deal with that (2^2 * 32)-4 term. This is where the power of a product rule comes into play. This rule states that (a * b)^n = a^n * b^n. In simple terms, if we have a product raised to a power, we can distribute the power to each factor in the product.

So, applying this rule to (2^2 * 32)-4, we get:

(22)-4 * (32)-4

Now, we need to simplify further using another exponent rule: the power of a power rule, which states that (am)n = a^(m*n). This means we multiply the exponents.

Applying this rule, we get:

2^(2 * -4) * 3^(2 * -4)

Which simplifies to:

2^-8 * 3^-8

Awesome! We've successfully simplified that complex term. Remember, understanding these exponent rules is crucial for tackling these types of problems. It might seem like a lot of rules to remember, but with practice, they'll become second nature.

Step 3: Putting It All Together

Now, let's put everything back into our main expression:

2^7 * (1 / (2^2)) * 3^5 * 2^-8 * 3^-8

To make things even clearer, let's rewrite 1 / (2^2) as 2^-2:

2^7 * 2^-2 * 3^5 * 2^-8 * 3^-8

Now, we can use the product of powers rule, which states that a^m * a^n = a^(m+n). This means when we multiply numbers with the same base, we add the exponents. Let's group the terms with the same base:

(2^7 * 2^-2 * 2^-8) * (3^5 * 3^-8)

Now, let's add the exponents:

2^(7 + (-2) + (-8)) * 3^(5 + (-8))

This simplifies to:

2^-3 * 3^-3

We're almost there! Remember, negative exponents mean we take the reciprocal. So:

(1 / (2^3)) * (1 / (3^3))

Step 4: Final Simplification

Now, let's calculate 2^3 and 3^3:

  • 2^3 = 2 * 2 * 2 = 8
  • 3^3 = 3 * 3 * 3 = 27

So, our expression becomes:

(1 / 8) * (1 / 27)

Finally, let's multiply the fractions:

1 / (8 * 27)

1 / 216

And there you have it! Our simplified expression is 1/216. Isn't that satisfying? We took a complex-looking expression and, by breaking it down step by step and applying the rules of exponents, we arrived at a simple fraction.

Key Takeaways

So, what did we learn today? Let's recap the key takeaways:

  1. Break it Down: When faced with a complex expression, break it down into smaller, more manageable parts.
  2. Prime Factorization: Express numbers in terms of their prime factors. This is especially helpful when dealing with exponents.
  3. Exponent Rules: Master the exponent rules (power of a product, power of a power, product of powers). These are your best friends when simplifying expressions.
  4. Negative Exponents: Remember that a negative exponent means taking the reciprocal.
  5. Step-by-Step: Work through the problem step by step, showing your work. This helps prevent errors and makes it easier to follow your logic.

Practice Makes Perfect

Simplifying exponential expressions might seem tricky at first, but the more you practice, the easier it will become. Try tackling similar problems, and don't be afraid to make mistakes. Mistakes are just learning opportunities in disguise! So, keep practicing, and you'll become a math whiz in no time.

Remember, guys, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. So, keep exploring, keep questioning, and keep having fun with math!