Simplifying Algebraic Expressions: A UFMG Math Challenge
Hey guys! Today, we're diving into a fun math problem from the UFMG (Universidade Federal de Minas Gerais), a renowned university in Brazil. We're going to tackle an algebraic expression simplification. This is a classic type of question you might see in a high school or even a college-level math exam. The goal is to take a complex-looking expression and reduce it to its simplest form. Let's break down the problem step-by-step to make sure we understand all the algebra and not just get the right answer, but why the answer is correct.
The Core Problem: Unpacking the Expression
Our expression is: a - 19 ⋅ (a - 13)^2 - a^2 ÷ (-1/a)^2, with the crucial condition that a ≠0. This condition is super important because it tells us that 'a' can't be zero, as division by zero is undefined in mathematics. This is one of those subtle things that often trips people up! Before we start, let's look at the options:
-a - 7√9a5√9-a5√9a7√9-a8√9
Now, let's get down to the actual calculation. Simplifying this expression involves a few key algebraic manipulations: expanding squared terms, dealing with exponents, and performing operations in the correct order (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Let's go!
Step-by-Step Simplification
First, let's tackle (a - 13)^2. This is where we need to remember the formula for squaring a binomial: (x - y)^2 = x^2 - 2xy + y^2. Applying this to our problem, we get:
(a - 13)^2 = a^2 - 2 â‹… a â‹… 13 + 13^2 = a^2 - 26a + 169
Now we have: a - 19 ⋅ (a^2 - 26a + 169) - a^2 ÷ (-1/a)^2
Next, let's address the term (-1/a)^2. When you square a fraction, you square both the numerator and the denominator. Also, remember that a negative number squared becomes positive. Thus:
(-1/a)^2 = (-1)^2 / a^2 = 1 / a^2
Our expression now looks like this: a - 19 ⋅ (a^2 - 26a + 169) - a^2 ÷ (1 / a^2)
Now, we multiply the -19 with the expanded square and deal with the division. Remember that dividing by a fraction is the same as multiplying by its reciprocal:
a - 19a^2 + 494a - 3211 - a^2 * a^2
Simplifying further:
a - 19a^2 + 494a - 3211 - a^4
Finally, re-arranging the terms and combining like terms:
-a^4 - 19a^2 + 495a - 3211
Looking at the options, none of them match exactly. However, we may have made a mistake in the calculations. Let's go through it one more time carefully.
Re-evaluating the Expression
Let's get back to the core expression: a - 19 ⋅ (a - 13)^2 - a^2 ÷ (-1/a)^2
- (a - 13)^2: Correctly expanded as
a^2 - 26a + 169. - (-1/a)^2: Correctly simplified to
1/a^2. - a - 19 â‹… (a^2 - 26a + 169):
a - 19a^2 + 494a - 3211 - - a^2 ÷ (1/a^2):
- a^2 * a^2 = -a^4
Putting it all together: a - 19a^2 + 494a - 3211 - a^4
Combining like terms (the 'a' terms): 495a
Re-arranging the terms and combining like terms:
-a^4 - 19a^2 + 495a - 3211
There seems to be an error in the given options or in the original question itself, as none of the provided choices match the simplified form of the expression. Given the options available, it's not possible to select a correct answer from the provided list. This is a good lesson, to ensure that you are correct, if you can not find a correct answer. Perhaps the original question had a typo or the options were misprinted!
Key Takeaways
- Order of Operations: Always follow PEMDAS/BODMAS to ensure correct calculations. This is absolutely critical.
- Binomial Expansion: Know how to expand expressions like
(x - y)^2. This comes up all the time. - Fractional Exponents: Understand how to handle exponents with fractions. Remember that negative numbers squared become positive. The most common mistakes are related to sign errors and exponent manipulation.
- Careful Calculation: Double-check your work, particularly when dealing with multiple steps. It's really easy to make small mistakes, so take it slow and steady.
Where to Go From Here?
Keep practicing these types of problems! The more you work through them, the more familiar you'll become with the techniques. Look for similar problems in textbooks, online, or in practice exams. Focus on understanding why you're doing each step, not just memorizing formulas. Try to create your own variations of the problem to test your understanding. Also, you might want to look at more questions from the UFMG. Good luck!
This UFMG question is a great example of how mathematical concepts are interconnected. It touches on basic algebra and requires careful attention to detail. Keep practicing, and you'll become a pro at these problems in no time! Keep learning, keep exploring, and enjoy the beauty of mathematics! Let me know if you want to try another problem. I'm always happy to help! And remember, practice makes perfect! So, keep at it, and you'll ace these algebra problems in no time! Don't get discouraged if it takes a bit to click; everyone learns at their own pace. Consistency is key!