Simple Math: Solve These Algebraic Expressions

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Simple Math: Solve These Algebraic Expressions

Hey guys! Ever found yourself staring at a math problem and thinking, "Wait, what am I supposed to do here?" Don't worry, you're not alone. Today, we're diving into some super straightforward algebraic expressions that will make you feel like a math whiz in no time. We've got two problems, (a) and (b), that are perfect for flexing those mathematical muscles. Whether you're a student trying to nail your homework or just someone who enjoys a good brain teaser, these will be a piece of cake. Let's break them down step by step and make sure everyone gets it. We'll be tackling problems involving combining like terms, a fundamental concept in algebra. So, grab your favorite beverage, get comfortable, and let's get solving!

Understanding the Basics: Like Terms

Before we jump into solving, let's quickly chat about like terms. In algebra, like terms are terms that have the exact same variable(s) raised to the exact same power(s). Think of them as buddies that can hang out and be combined. For example, in an expression like 3x + 5x - 2x, all the terms have the variable x raised to the power of 1 (which we usually don't write). Because they all have x, they are like terms. This means we can add or subtract their coefficients (the numbers in front of the variable). So, 3x + 5x - 2x would become (3 + 5 - 2)x, which simplifies to 6x. Easy peasy, right? The key takeaway here is that you can only combine terms if they share the same variable and exponent. If you have 3x + 5y, you can't combine them because one has an x and the other has a y. They aren't like terms! This concept is going to be super important for the problems we're about to tackle. We'll see how it applies directly to our expressions.

Problem (a): Simplifying the First Expression

Alright, let's get down to business with our first problem: 3x⁴ + 2x⁴ + 12x⁴ - 2x⁴. Take a good look at it. What do you notice? That's right, every single term has the variable x raised to the power of 4 (written as x⁴). This means all these terms are like terms! This is fantastic news because it means we can simply combine the coefficients. The coefficients are the numbers in front of x⁴: 3, +2, +12, and -2. So, to solve this, we just need to perform the arithmetic on these numbers: 3 + 2 + 12 - 2. Let's do it together. First, 3 + 2 gives us 5. Then, 5 + 12 equals 17. Finally, we subtract 2 from 17, which leaves us with 15. So, the simplified expression is 15x⁴. See? You just combined four terms into one! This is the power of recognizing like terms. It takes a potentially confusing string of numbers and variables and boils it down to something much cleaner and easier to understand. When you see expressions like this, always do a quick scan to identify if the variables and their exponents match up. If they do, you're golden and can proceed with combining the coefficients. Remember, the variable part (x⁴ in this case) stays the same; only the numerical coefficients change during the addition or subtraction.

Step-by-Step Solution for (a)

To make it crystal clear, let's lay out the steps for problem (a):

  1. Identify Like Terms: Look at the expression 3x⁴ + 2x⁴ + 12x⁴ - 2x⁴. All terms contain the variable x raised to the power of 4 (x⁴). Therefore, they are all like terms.
  2. Isolate Coefficients: The coefficients are the numbers multiplying x⁴: 3, 2, 12, and -2.
  3. Perform Arithmetic Operations: Add and subtract the coefficients according to the signs in the expression: 3 + 2 + 12 - 2.
    • 3 + 2 = 5
    • 5 + 12 = 17
    • 17 - 2 = 15
  4. Combine with the Variable: Attach the resulting coefficient to the common variable term. Since the common term is x⁴, the final answer is 15x⁴.

This systematic approach ensures that you don't miss any steps and that you correctly combine the terms. It’s a fundamental skill that builds a strong foundation for more complex algebraic manipulations down the line. Keep practicing this, and you'll be a master of combining like terms in no time!

Problem (b): Tackling the Second Expression

Now, let's move on to our second challenge: 5x⁵ + 10x⁵ - 15x⁵ + 3x⁵. Just like with the first problem, take a moment to examine this expression. What do you notice about the terms? You've probably guessed it – they all have the variable x raised to the power of 5 (or x⁵). That means, once again, we're dealing with a situation where all terms are like terms. This is great! It signifies that we can proceed with combining the coefficients. The coefficients here are 5, +10, -15, and +3. Our task now is to perform the arithmetic operation on these numbers: 5 + 10 - 15 + 3. Let's work through it. Starting from the left, 5 + 10 gives us 15. Next, we subtract 15 from 15, which equals 0. Finally, we add 3 to 0, resulting in 3. Therefore, the simplified expression is 3x⁵. How cool is that? We simplified another expression by just adding and subtracting the coefficients of the like terms. This highlights how crucial it is to spot these like terms. Without recognizing them, you might be tempted to think these problems are much harder than they actually are. The consistency in the variable part (x⁵) is the key that unlocks the simplification.

Step-by-Step Solution for (b)

Let's break down the solution for problem (b) step-by-step, just like we did for problem (a):

  1. Identify Like Terms: Examine the expression 5x⁵ + 10x⁵ - 15x⁵ + 3x⁵. All terms contain the variable x raised to the power of 5 (x⁵). Hence, they are all like terms.
  2. Isolate Coefficients: The coefficients are the numerical values multiplying x⁵: 5, 10, -15, and 3.
  3. Perform Arithmetic Operations: Add and subtract the coefficients based on the operators in the expression: 5 + 10 - 15 + 3.
    • 5 + 10 = 15
    • 15 - 15 = 0
    • 0 + 3 = 3
  4. Combine with the Variable: Append the calculated coefficient to the common variable term. Since the common term is x⁵, the final answer is 3x⁵.

By following these structured steps, you can confidently solve such problems. The process is straightforward and focuses on the fundamental algebraic concept of combining like terms. This method is applicable to any expression where like terms are present, making it a versatile skill in your mathematical toolkit.

Why is this Important, Guys?

So, why do we even bother with this stuff? Understanding how to combine like terms is a foundational skill in algebra. It's like learning your ABCs before you can write a novel. This skill is essential for simplifying expressions, which is the first step in solving more complex equations. Imagine trying to solve (3x + 2) + (5x - 1) = 10 without knowing how to combine 3x and 5x into 8x, or 2 and -1 into 1. It would be way harder! Simplifying makes equations tidier and easier to manage. Furthermore, mastering this concept helps build your confidence in tackling more advanced mathematical topics, like polynomial operations, factoring, and graphing. It’s about making math less intimidating and more accessible. So, the next time you see an algebraic expression, take a deep breath, look for those like terms, and simplify away! You've got this, and it's a crucial step in your math journey.

Conclusion

We've successfully navigated through two algebraic expressions, simplifying them by combining like terms. For problem (a), 3x⁴ + 2x⁴ + 12x⁴ - 2x⁴, we found the answer to be 15x⁴. And for problem (b), 5x⁵ + 10x⁵ - 15x⁵ + 3x⁵, the simplified form is 3x⁵. Remember, the key to solving these problems quickly and accurately is to identify terms with the same variable and the same exponent, and then simply perform the arithmetic on their coefficients. This fundamental skill is your gateway to understanding more complex algebraic concepts. Keep practicing, and don't hesitate to break down problems step-by-step. Happy solving, math adventurers!