Math Subtraction Problems: Step-by-Step Solutions
Hey guys! Today, we're diving into some math subtraction problems. We've got a couple of juicy algebraic expressions to tackle. Don't worry, we'll break it down step by step so it's super easy to follow. If you've ever felt a little intimidated by subtracting polynomials, this is the place to be. We'll transform those math woes into "Aha!" moments. So, let's jump right in and get those subtraction skills sharpened!
Problem a: Subtract -2m⁴n³ + 12 from 18m⁴n³ - 6m³n² + 6
Okay, let's start with the first part of our adventure: subtracting -2m⁴n³ + 12 from 18m⁴n³ - 6m³n² + 6. The key here is to remember that we're not just subtracting numbers, but entire algebraic expressions. This means we need to pay close attention to the signs and the like terms.
Understanding the Basics
First off, let's quickly recap what "like terms" are. Like terms are terms that have the same variables raised to the same powers. For example, 3x² and -5x² are like terms because they both have x². However, 3x² and 3x³ are not like terms because the exponents are different. Recognizing like terms is crucial because we can only combine (add or subtract) like terms.
When we subtract one expression from another, it’s like we’re distributing a negative sign across the entire expression we’re subtracting. It's super important to get this step right, or else the whole problem goes sideways!
Setting Up the Subtraction
So, we need to subtract -2m⁴n³ + 12 from 18m⁴n³ - 6m³n² + 6. This translates to:
(18m⁴n³ - 6m³n² + 6) - (-2m⁴n³ + 12)
See how we've put the entire second expression in parentheses? That’s because we're subtracting the whole thing, not just the first term.
Distributing the Negative Sign
Now comes the crucial part – distributing that negative sign. It's like giving a little minus sign to everyone inside the second set of parentheses. This changes the signs of each term:
18m⁴n³ - 6m³n² + 6 + 2m⁴n³ - 12
Notice how -2m⁴n³ became +2m⁴n³ and +12 became -12? That's the magic of the negative sign distribution!
Combining Like Terms
Next up, we need to identify and combine those like terms. Let’s group them together:
(18m⁴n³ + 2m⁴n³) - 6m³n² + (6 - 12)
Now we can add or subtract the coefficients (the numbers in front of the variables) of the like terms:
18m⁴n³ + 2m⁴n³ = 20m⁴n³-6m³n²has no like terms, so it stays as is.6 - 12 = -6
The Final Answer for Problem a
Putting it all together, we get our final answer:
20m⁴n³ - 6m³n² - 6
And that’s it! We’ve successfully subtracted the first expression from the second. See? It's not so scary when you break it down step by step.
Problem b: Subtract 12/5 x³ - 7x² + 2 from x⁴ + 7/10 x³ + 1/2 x² - 5/3
Alright, guys, let's move on to the second subtraction problem! This one involves fractions, which might seem a little intimidating, but trust me, we've got this. We need to subtract 12/5 x³ - 7x² + 2 from x⁴ + 7/10 x³ + 1/2 x² - 5/3. Just like before, we’ll take it one step at a time to make sure we don’t miss anything.
Setting Up the Subtraction
First, let's write out the subtraction expression:
(x⁴ + 7/10 x³ + 1/2 x² - 5/3) - (12/5 x³ - 7x² + 2)
Remember those parentheses? They’re super important! We're subtracting the entire second expression, so we need to keep it grouped together initially.
Distributing the Negative Sign (Again!)
Just like before, our next step is to distribute the negative sign across the second set of parentheses. This means changing the sign of each term inside:
x⁴ + 7/10 x³ + 1/2 x² - 5/3 - 12/5 x³ + 7x² - 2
Notice how 12/5 x³ became -12/5 x³, -7x² became +7x², and +2 became -2? We're on a roll!
Identifying and Grouping Like Terms
Now, let's identify and group our like terms. This is where things can get a little tricky with the fractions, but we’ll tackle it together:
x⁴ + (7/10 x³ - 12/5 x³) + (1/2 x² + 7x²) + (-5/3 - 2)
We've grouped the x³ terms together, the x² terms together, and the constants together. The x⁴ term is all alone, so it'll just tag along for now.
Combining Like Terms: Fractions Edition!
Here’s where we put on our fraction-combining hats. Remember, to add or subtract fractions, they need to have a common denominator.
Combining the x³ Terms
Let's start with the x³ terms: 7/10 x³ - 12/5 x³. We need to find a common denominator for 10 and 5, which is 10. So, we'll rewrite 12/5 as an equivalent fraction with a denominator of 10:
12/5 = (12 * 2) / (5 * 2) = 24/10
Now we can rewrite our expression:
7/10 x³ - 24/10 x³ = (7 - 24) / 10 x³ = -17/10 x³
Combining the x² Terms
Next up, the x² terms: 1/2 x² + 7x². We can think of 7x² as 7/1 x². We need a common denominator for 2 and 1, which is 2. So, we'll rewrite 7/1 as an equivalent fraction with a denominator of 2:
7/1 = (7 * 2) / (1 * 2) = 14/2
Now we can add:
1/2 x² + 14/2 x² = (1 + 14) / 2 x² = 15/2 x²
Combining the Constants
Finally, let’s combine the constants: -5/3 - 2. We can think of -2 as -2/1. We need a common denominator for 3 and 1, which is 3. So, we'll rewrite -2/1 as an equivalent fraction with a denominator of 3:
-2/1 = (-2 * 3) / (1 * 3) = -6/3
Now we can subtract:
-5/3 - 6/3 = (-5 - 6) / 3 = -11/3
The Final Answer for Problem b
Putting it all together, we get our final answer:
x⁴ - 17/10 x³ + 15/2 x² - 11/3
Woohoo! We did it! We tackled fractions and variables, and came out on top. Give yourself a pat on the back; you deserve it!
Final Thoughts
So, guys, that’s how you conquer subtraction problems with algebraic expressions. Remember the key takeaways: distribute the negative sign carefully, identify and group those like terms, and take your time with the fractions. Math might seem like a monster sometimes, but breaking it down into smaller steps makes it way less scary. Keep practicing, and you’ll be subtracting like a pro in no time! And hey, if you ever get stuck, just remember this guide, and you’ll be all set. Happy calculating!