Hey guys! Let's dive into a fun math problem today. We're going to break down the expression (12⁻⁵)² and figure out which of the given options is equivalent. Math can seem intimidating at first, but with a step-by-step approach, it becomes much easier to handle. We'll go through the fundamental rules of exponents and apply them to solve this problem. So, buckle up and let's get started!
Understanding the Problem
First, let's rewrite the question to make sure we're crystal clear on what we need to do. The core of our task lies in simplifying the expression (12⁻⁵)². To effectively tackle this, we'll need to recall some key exponent rules. Remember, exponents are a shorthand way of showing repeated multiplication. A negative exponent indicates a reciprocal, and when we raise a power to another power, we multiply the exponents. Now, let’s consider the options presented:
- 12³ ⋅ 12⁻¹
- 12⁻³
- 12⁹ / 12⁻³
- 1 / 12¹⁰
Our mission is to find which of these options, when simplified, matches the simplified form of (12⁻⁵)². To do this, we'll first simplify the original expression and then compare it to each of the options. This involves applying the power of a power rule and understanding how negative exponents work. We’ll take it slow and make sure we understand each step. Math is like building blocks – each concept builds on the previous one, so getting the basics right is super important.
Key Exponent Rules
Before we dive into the solution, let's quickly recap the exponent rules we'll be using. These rules are the foundation for simplifying expressions with exponents, and mastering them will make these problems much easier. There are three main rules that come into play here:
- Power of a Power Rule: (aᵐ)ⁿ = aᵐⁿ
- Product of Powers Rule: aᵐ ⋅ aⁿ = aᵐ⁺ⁿ
- Quotient of Powers Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
- Negative Exponent Rule: a⁻ⁿ = 1 / aⁿ
The power of a power rule tells us that when we raise a power to another power, we multiply the exponents. For example, (x²)³ = x²*³ = x⁶. This rule is crucial for simplifying our initial expression. The product of powers rule states that when we multiply powers with the same base, we add the exponents. For instance, x² ⋅ x³ = x²⁺³ = x⁵. This will be useful for simplifying some of the options. The quotient of powers rule says that when we divide powers with the same base, we subtract the exponents. So, x⁵ / x² = x⁵⁻² = x³. Lastly, the negative exponent rule tells us that a negative exponent means we take the reciprocal of the base raised to the positive exponent. For example, x⁻² = 1 / x². This is essential for dealing with negative exponents in our options.
With these rules in mind, we’re well-equipped to tackle the problem. Remember, practice makes perfect, so the more you work with these rules, the more natural they will become.
Step-by-Step Solution
Okay, let's tackle this step by step. Our initial expression is (12⁻⁵)². The first thing we need to do is apply the power of a power rule. This rule states that (aᵐ)ⁿ = aᵐⁿ. In our case, a = 12, m = -5, and n = 2. So, we have:
(12⁻⁵)² = 12⁻⁵ * ² = 12⁻¹⁰
Now we have simplified the original expression to 12⁻¹⁰. Next, we can use the negative exponent rule to rewrite this. The negative exponent rule states that a⁻ⁿ = 1 / aⁿ. Applying this rule, we get:
12⁻¹⁰ = 1 / 12¹⁰
So, the simplified form of our original expression is 1 / 12¹⁰. Now, let's examine the given options and see which one matches this simplified form. This is a crucial step because it confirms whether our initial simplification was correct and helps us identify the equivalent expression.
Evaluating the Options
Now let's evaluate each of the given options to see which one is equivalent to 1 / 12¹⁰. This is where we put our understanding of exponent rules to the test. We'll simplify each option step by step and compare the result to our simplified expression.
Option 1: 12³ ⋅ 12⁻¹
To simplify this, we use the product of powers rule, which states that aᵐ ⋅ aⁿ = aᵐ⁺ⁿ. So, we have:
12³ ⋅ 12⁻¹ = 12³ ⁺ ⁽⁻¹⁾ = 12²
12² is not equal to 1 / 12¹⁰, so this option is incorrect.
Option 2: 12⁻³
This expression is already in a simplified form. Using the negative exponent rule, we can rewrite it as:
12⁻³ = 1 / 12³
1 / 12³ is not equal to 1 / 12¹⁰, so this option is also incorrect.
Option 3: 12⁹ / 12⁻³
To simplify this, we use the quotient of powers rule, which states that aᵐ / aⁿ = aᵐ⁻ⁿ. So, we have:
12⁹ / 12⁻³ = 12⁹ ⁻ ⁽⁻³⁾ = 12⁹ ⁺ ³ = 12¹²
12¹² is not equal to 1 / 12¹⁰, making this option incorrect as well.
Option 4: 1 / 12¹⁰
This expression is already in its simplest form and it directly matches our simplified expression. Therefore, this option is the correct one.
The Correct Answer
After evaluating all the options, we can confidently say that the expression equivalent to (12⁻⁵)² is 1 / 12¹⁰. This matches option 4, which was already in its simplest form and directly equivalent to our simplified original expression. It’s always a good feeling when the answer is clear and straightforward! We arrived at this answer by systematically applying the exponent rules, showing the power of a methodical approach in math problems.
Conclusion
So, guys, we've successfully navigated this math problem! We started with the expression (12⁻⁵)², simplified it using the power of a power and negative exponent rules, and then compared it to the given options. We found that 1 / 12¹⁰ is the equivalent expression. The key takeaways here are the importance of understanding and applying exponent rules, and the value of breaking down a problem into manageable steps. Math might seem daunting at times, but with a clear strategy and a little practice, you can conquer it. Keep up the great work, and remember, every problem solved is a step forward in your math journey! Always feel free to revisit these steps and rules as you tackle similar problems. Happy solving!