Tamika's Math Error: A Step-by-Step Analysis

Hey guys! Let's dive into Tamika's math problem and figure out where things went south. We're going to break down her work step by step, pinpoint the exact error, and learn a thing or two about simplifying exponents in the process. It's like a math detective story, and we're the investigators! The original problem is: $\begin{aligned} \frac{18 a^{-5} b^{-6}}{30 a^3 b^{-5}} & =\frac{3 a^{-2} b^{-11}}{5} \ & =\frac{3}{5 a^2 b^{11}} \end{aligned}$ This looks like a classic exponent simplification problem, but something went wrong along the way. Let's get to the bottom of it.

Decoding the Expression: The Initial Steps

Alright, first things first, let's look at the original expression: $\frac{18 a^{-5} b^{-6}}{30 a^3 b^{-5}}$. The goal here is to simplify this fraction as much as possible. This involves dealing with the coefficients (the numbers), and the variables with their exponents. Initially, Tamika attempts to simplify this equation. The first thing she does is divide the coefficients. The initial coefficient in the numerator is 18 and the initial coefficient in the denominator is 30. Both numbers can be divided by 6, which is their greatest common factor (GCF). 18 divided by 6 is 3, and 30 divided by 6 is 5. So, the coefficients were handled correctly. The original expression can now be written as $\frac{3 a^{-5} b^{-6}}{5 a^3 b^{-5}}$. Now it's time to deal with the variables and their exponents! But before we go any further, let's remember a crucial rule of exponents: When dividing terms with the same base, you subtract the exponents. That's the secret sauce here.

So, when dealing with the variable a, we have a raised to the power of -5 in the numerator and a raised to the power of 3 in the denominator. To simplify, we should subtract the exponents: -5 - 3 = -8. Thus, the correct term for a should be $a^{-8}$. Now, let's talk about the variable b. In the numerator, b is raised to the power of -6, and in the denominator, b is raised to the power of -5. Again, we subtract the exponents: -6 - (-5) = -6 + 5 = -1. So the correct term for b should be $b^{-1}$.

The Breakdown: Identifying Tamika's Error

Now, let's analyze Tamika's moves. She appears to have made a mistake in the first step. When she simplifies the terms with the variable a, she seems to have added the exponents instead of subtracting them. Her solution showed the variable a with the power of -2, when in fact, the correct result should be -8. Remember, when you divide exponential terms, you subtract the exponents. In her work, she combined $a^{-5}$ and $a^3$. Instead of subtracting, she seems to have combined the terms incorrectly. This leads to the first mistake in her work. Tamika did not correctly apply the exponent rules when working with the variable a. This is a critical misunderstanding of how exponents behave in division. She should have subtracted the exponent in the denominator from the exponent in the numerator.

Let's focus on the variable b. Tamika shows that the power of b is -11. However, the correct answer for the exponent of b should be -1. This means Tamika also made a mistake while handling the variable b. This mistake can be explained by the fact that she may have added exponents instead of subtracting them. Always double-check your sign when working with negative exponents! This is where the minus signs and all the negative numbers come into play, and it's easy to get mixed up. Remember that a negative exponent in the denominator becomes positive in the numerator and vice-versa. Always keep the exponent rules at the top of your head, as the core rule when handling the variables. So, to wrap it up, the correct simplification should look like this: $\frac{18 a^{-5} b^{-6}}{30 a^3 b^{-5}} = \frac{3 a^{-8} b^{-1}}{5} = \frac{3}{5 a^8 b}$

Correcting the Course: The Right Way to Simplify

So, to get it right, here's the correct way to solve the problem. First, we simplify the coefficients by dividing both the numerator and the denominator by their GCF, which is 6. This leaves us with $\frac{3 a^{-5} b^{-6}}{5 a^3 b^{-5}}$. Then, we handle the variables. For a, subtract the exponents: -5 - 3 = -8, giving us $a^{-8}$. For b, subtract the exponents: -6 - (-5) = -6 + 5 = -1, giving us $b^{-1}$. This leaves us with $\frac{3 a^{-8} b^{-1}}{5}$. Lastly, to make the exponents positive, we move $a^{-8}$ and $b^{-1}$ to the denominator, which gives us $\frac{3}{5 a^8 b}$. This is the simplified form of the original expression. The original problem provided can be solved if we apply the rules correctly and keep track of our signs.

In essence, the key to avoiding Tamika's mistake is to remember the rules of exponent division. Always subtract the exponents when dividing terms with the same base. Make sure to double-check your signs, especially when dealing with negative exponents. Break the problem down step by step and work carefully, and you'll be golden! Remember, practice makes perfect, and with a little effort, you'll be simplifying exponents like a pro.

Conclusion: Lessons Learned

So, what have we learned, guys? Tamika's error stemmed from a misunderstanding of the rules of exponents. She either added exponents when she should have subtracted them, or she made mistakes when handling negative numbers. The main takeaway is to carefully apply the exponent rules and double-check your work, paying close attention to the signs. Always remember: when dividing exponential terms, you subtract the exponents of the same base. By practicing these rules and being mindful of the signs, you can avoid making the same mistakes and master the art of simplifying expressions with exponents. Keep up the good work and keep learning!