Tamika's Exponent Error: A Math Mystery Solved
Hey guys! Today, we're diving into a common algebra hiccup by dissecting a problem Tamika tackled. We'll figure out where she went wrong in simplifying an expression with exponents. It's a learning journey, so let's put on our math detective hats and get started!
The Problem: A Step-by-Step Breakdown
Tamika's task was to simplify this expression:
She arrived at the following solution:
Our mission, should we choose to accept it (and we do!), is to pinpoint the exact step where Tamika went astray. Let's break it down and see where the mathematical misstep occurred.
Initial Simplification: Numbers First
Okay, so let's start by focusing on those coefficients, the numbers hanging out in front of our variables. We have 18 in the numerator (that's the top part of the fraction) and 30 in the denominator (the bottom part). We need to simplify this fraction. What's the greatest common factor of 18 and 30? If you said 6, you're on fire!
Dividing both 18 and 30 by 6, we get:
So far, so good! Tamika nailed this first part. We've got the numerical part of our simplified expression, which is 3/5. This is a crucial first step, so we're building a solid foundation for the rest of the problem. It's like laying the groundwork for a skyscraper – you need a strong base to build something amazing! Remember, math is all about precision, and getting these initial simplifications right is key to avoiding errors down the line.
Diving into the Variables: The 'a' Exponents
Now, let's turn our attention to the 'a' variables and their exponents. This is where things can get a little tricky, but don't worry, we'll tackle it together! We have a⁻⁵ in the numerator and a³ in the denominator. Remember the rule for dividing exponents with the same base? We subtract the exponent in the denominator from the exponent in the numerator. This is a fundamental rule of exponents, and it's essential for simplifying expressions like this. Think of it like this: when you divide, you're essentially "canceling out" some of the factors. In the case of exponents, this "canceling out" translates to subtraction.
So, we have:
a⁻⁵ / a³ = a⁽⁻⁵⁻³⁾
Now, let's do the math: -5 - 3 = -8. Therefore:
a⁻⁵ / a³ = a⁻⁸
Tamika, however, wrote a⁻². This is where the first red flag pops up! It seems like she might have added the exponents instead of subtracting them. This is a common mistake, so don't feel bad if you've made it too! The key is to remember the rules and apply them carefully. We're here to learn from these mistakes and become math masters! So, we've identified a potential error, and we're one step closer to solving the puzzle.
Spotting the 'b' Exponent Error
Alright, let's shift our focus to the 'b' variables and their exponents. This is where we'll either confirm our suspicions or uncover another layer to Tamika's mathematical adventure! We've got b⁻⁶ in the numerator and b⁻⁵ in the denominator. Just like with the 'a' variables, we need to remember the rule for dividing exponents with the same base: subtract the exponent in the denominator from the exponent in the numerator.
So, we have:
b⁻⁶ / b⁻⁵ = b⁽⁻⁶⁻⁽⁻⁵⁾⁾
Be careful with those negative signs! Subtracting a negative number is the same as adding its positive counterpart. So, -6 - (-5) becomes -6 + 5, which equals -1. Therefore:
b⁻⁶ / b⁻⁵ = b⁻¹
Now, let's compare this to what Tamika wrote. She has b⁻¹¹. Whoa, that's a big difference! It looks like she might have made a similar mistake here as with the 'a' exponents, perhaps adding instead of subtracting, or maybe there was a sign error. Whatever the cause, this confirms that there's definitely an error in how she handled the 'b' exponents. We're piecing together the puzzle, and it's becoming clear where the missteps occurred. Identifying these errors is a crucial part of the learning process, and we're doing a great job so far!
Putting It All Together: The Correct Simplification
Okay, we've dissected Tamika's work piece by piece, and now it's time to assemble the correct solution. We've already simplified the numerical part (18/30) to 3/5. We've also correctly simplified the 'a' exponents to a⁻⁸ and the 'b' exponents to b⁻¹. Let's put it all together:
But wait, there's more! We're not quite done yet. Remember, we want to express our answer with positive exponents whenever possible. This is a standard practice in simplifying expressions, as it makes them easier to understand and work with. To get rid of the negative exponents, we need to move the terms with negative exponents to the opposite side of the fraction. If a term with a negative exponent is in the numerator, we move it to the denominator, and vice versa.
So, a⁻⁸ in the numerator becomes a⁸ in the denominator, and b⁻¹ in the numerator becomes b¹ (or simply b) in the denominator. This gives us:
Ta-da! We've arrived at the correctly simplified expression. It's like reaching the summit of a mathematical mountain – the view is pretty great, right? We've not only solved the problem but also reinforced our understanding of exponent rules. This is the power of working through problems step by step and identifying potential pitfalls. We're not just memorizing rules; we're building a solid foundation of understanding.
Identifying Tamika's Error: The Final Verdict
So, after our thorough investigation, we've pinpointed Tamika's error. Let's recap the original expression and Tamika's steps:
Tamika's solution:
We discovered that Tamika made a mistake when simplifying the exponents. Specifically, she added the exponents instead of subtracting them when dividing terms with the same base. This led to incorrect exponents for both 'a' and 'b'. This is a classic exponent blunder, and it highlights the importance of carefully applying the rules of exponents. It's like a chef forgetting a key ingredient – the dish just won't turn out quite right! In this case, the "ingredient" was the correct application of the exponent rules.
Key Takeaways: Mastering Exponents
Alright, guys, we've reached the end of our mathematical journey, and what a journey it has been! We've not only identified Tamika's error but also reinforced some crucial concepts about exponents. Let's recap the key takeaways from this problem:
- The Division Rule for Exponents: When dividing terms with the same base, you subtract the exponent in the denominator from the exponent in the numerator. This is a fundamental rule, and it's essential to have it locked down. Think of it as the golden rule of exponent division!
- Negative Exponents: Remember that a term with a negative exponent can be rewritten by moving it to the opposite side of the fraction (numerator to denominator or vice versa) and changing the sign of the exponent. This is like a mathematical makeover – we're just changing the appearance of the expression without changing its value.
- Step-by-Step Simplification: Breaking down complex problems into smaller, manageable steps is a powerful strategy. It allows you to focus on each part individually and reduces the chance of making errors. It's like building a house brick by brick – each step is important, and the final result is a solid structure.
- Double-Checking Your Work: Always take the time to review your work and ensure you've applied the rules correctly. It's like proofreading a document before submitting it – a little extra effort can catch those sneaky errors.
By understanding these key concepts and practicing regularly, you'll be well on your way to mastering exponents and tackling even the most challenging algebraic expressions. Keep up the awesome work, and remember, every mistake is a learning opportunity!
So, the final answer to the question, "What is Tamika's error?" is:
A. She added the exponents.
