Solving The Expression: (-2 A^4)(2 A^2)(-5 A)

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Hey guys! Let's dive into solving this math expression together. We've got (-2 a^4)(2 a^2)(-5 a), and it might look a little intimidating at first, but trust me, we'll break it down step by step. Math can be like a puzzle, and it's super satisfying when you finally fit all the pieces together. So, grab your thinking caps, and let’s get started!

Understanding the Basics

Before we jump into the solution, let's quickly review some fundamental concepts. When we're dealing with expressions like this, we need to remember the rules of exponents and how to multiply coefficients. The coefficients are the numbers in front of the variables (like -2, 2, and -5 in our expression), and the exponents tell us how many times to multiply the variable by itself (like the 4 in a^4).

Exponents: Remember the rule: when multiplying terms with the same base (in our case, 'a'), you add the exponents. So, a^m * a^n = a^(m+n). This is super important for simplifying our expression. For example, a^2 * a^3 becomes a^(2+3) = a^5.

Coefficients: Multiplying coefficients is straightforward. Just multiply the numbers together. For example, -2 * 2 * -5 gives us a positive result because a negative times a negative is a positive. We’ll figure out the exact number in our steps below. Think of it like combining quantities – if you have -2 of something, then double it, and then multiply by -5, how much do you end up with?

Order of Operations: While in this particular problem the order doesn't dramatically change the outcome due to the associative property of multiplication, it's always good practice to remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). We're mainly dealing with multiplication here, so we’re good to go without worrying too much about other operations this time. But keep this in mind for more complex problems in the future!

Step-by-Step Solution

Okay, let’s break down the expression (-2 a^4)(2 a^2)(-5 a) step by step. This is where the magic happens, and we see how everything fits together. I promise it’s not as scary as it looks!

Step 1: Multiply the Coefficients

First, let's focus on the coefficients: -2, 2, and -5. We need to multiply these together. Let’s start with the first two:

-2 * 2 = -4

Now, let’s multiply that result by the last coefficient:

-4 * -5 = 20

So, the product of the coefficients is 20. Remember that a negative times a negative gives a positive, which is why we end up with a positive 20. Coefficients are just the numerical part of our terms, and getting this part right is crucial.

Step 2: Multiply the Variables

Next, we'll deal with the variables. We have a^4, a^2, and a (which is the same as a^1). Remember our rule for multiplying variables with exponents? We add the exponents together.

So, we have:

a^4 * a^2 * a^1

Adding the exponents:

4 + 2 + 1 = 7

This gives us a^7. It’s like we're combining the powers of 'a'. If you have 'a' multiplied by itself 4 times, then multiplied by itself 2 times, and then multiplied by itself 1 more time, you end up with 'a' multiplied by itself 7 times.

Step 3: Combine the Results

Now that we've multiplied the coefficients and the variables, let’s put it all together. We found that the coefficients multiply to 20, and the variables multiply to a^7.

So, our expression simplifies to:

20a^7

And that’s it! We’ve solved it. See? Not so scary after all.

Common Mistakes to Avoid

Everyone makes mistakes, and that’s totally okay! It’s part of learning. But being aware of common pitfalls can help you avoid them. Here are a few mistakes people often make when solving expressions like this:

  1. Forgetting the Exponent Rule: The most common mistake is forgetting to add the exponents when multiplying variables. Remember, a^m * a^n = a^(m+n), not a^(m*n). This is a super important rule, so make sure it’s solid in your mind.
  2. Coefficient Sign Errors: Pay close attention to the signs of the coefficients. A negative times a negative is a positive, and a negative times a positive is a negative. It’s easy to make a small mistake here, so double-check your work.
  3. Treating 'a' as 0: Sometimes, people might forget that 'a' by itself is the same as a^1. This can lead to incorrect exponent addition. Always remember that any variable without an explicit exponent has an exponent of 1.
  4. Incorrect Order of Operations: Although it wasn't a big issue in this problem, always keep PEMDAS/BODMAS in mind. If there were other operations involved (like addition or subtraction), you’d need to follow the correct order to get the right answer.

Practice Makes Perfect

The best way to get better at math is, you guessed it, practice! The more you practice, the more comfortable you'll become with these concepts. Try solving similar problems, and don’t be afraid to make mistakes. Mistakes are just learning opportunities in disguise.

Here are a few practice problems you can try:

  1. (3b2)(-4b3)(2b)
  2. (-5x5)(x2)(-3x^3)
  3. (4c3)(-2c)(5c4)

Solve these using the same steps we used above, and you’ll be a pro in no time!

Real-World Applications

You might be wondering,