Simplify Algebraic Expressions: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of simplifying algebraic expressions. This is a fundamental skill in mathematics, and mastering it will make your life so much easier when tackling more complex problems. We'll break down three expressions step-by-step, so you can confidently simplify them yourself.

a. Simplifying ${\frac{a \cdot 3 \cdot x + b}{2}}$

Let's kick things off with our first expression: ${\frac{a \cdot 3 \cdot x + b}{2}}$. Simplifying algebraic expressions like this involves a few key strategies, and in this case, the main goal is to see if we can further consolidate the terms in the numerator or potentially find a common factor to cancel out with the denominator. The first thing we can do is rewrite the expression to make it a bit cleaner and easier to read. We have the term ${a \cdot 3 \cdot x}$, which is just a fancy way of saying ${3ax}$. So, let’s rewrite our expression as:

${\frac{3ax + b}{2}}$

Now, looking at this, we need to ask ourselves: can we simplify this any further? Are there any common factors between the numerator and the denominator? In this case, the answer is no. The numerator has two terms: ${3ax}$ and ${b}$. These terms are unlike terms because one contains the variables ${a}$ and ${x}$, and the other is just the variable ${b}$. Unlike terms cannot be combined through addition or subtraction. The denominator is simply the number 2. There isn’t a common factor between ${3ax}$, ${b}$, and 2 that we can factor out and cancel. Therefore, this expression is already in its simplest form. We can't combine ${3ax}$ and ${b}$ because they are not like terms – one has variables ${a}$ and ${x}$, and the other is just ${b}$. Also, there's no common factor between the numerator (${3ax + b}$) and the denominator (2). Think of it like trying to add apples and oranges – they're just different! So, we're done here. This expression is as simple as it gets!

Key takeaway: Always look for like terms to combine and common factors to cancel out, but remember that sometimes an expression is already in its simplest form. Recognizing when to stop is just as important as knowing how to simplify.

b. Simplifying ${\frac{20m + 5n}{5}}$

Next up, let's tackle the expression ${\frac{20m + 5n}{5}}$. When simplifying algebraic expressions, especially fractions like this, the key is to look for common factors. In this case, we have a numerator with two terms, ${20m}$ and ${5n}$, and a denominator of 5. Our goal is to see if we can factor out a common factor from the numerator that we can then cancel with the denominator. Let's take a closer look at the numerator: ${20m + 5n}$. Both terms have a common factor of 5. We can factor out this 5, which means we rewrite the expression as a product of 5 and another expression. Here's how it looks:

${5(4m + n)}$

So, we’ve rewritten the numerator by factoring out the common factor of 5. Now, let's put this back into our original expression:

${\frac{5(4m + n)}{5}}$

Now we can see that we have a factor of 5 in both the numerator and the denominator. This is great news because we can cancel these out! When we cancel the 5 in the numerator with the 5 in the denominator, we’re left with:

${4m + n}$

And that’s it! We’ve successfully simplified the expression ${\frac{20m + 5n}{5}}$ to ${4m + n}$. This is the simplest form because there are no more common factors to cancel or like terms to combine. The variables ${m}$ and ${n}$ are different, so we can’t add them together. Remember, factoring out common factors is a crucial technique in simplifying expressions, and it’s something you’ll use again and again in algebra.

Key takeaway: Always be on the lookout for common factors. Factoring them out can significantly simplify your expressions and make them much easier to work with.

c. Simplifying ${25d^4 + 20d^3}$

Alright, let's move on to our third expression: ${25d^4 + 20d^3}$. This expression doesn't involve a fraction like the previous one, but the principle of simplifying algebraic expressions remains the same: we need to look for common factors. In this case, we're looking for common factors between the two terms, ${25d^4}$ and ${20d^3}$. This involves identifying both numerical and variable factors that are shared by both terms. Let's start with the numerical coefficients: 25 and 20. What's the greatest common factor (GCF) of 25 and 20? The factors of 25 are 1, 5, and 25. The factors of 20 are 1, 2, 4, 5, 10, and 20. The greatest common factor is 5. So, 5 is a common factor we can pull out. Now, let's look at the variable parts: ${d^4}$ and ${d^3}$. Remember that ${d^4}$ means ${d \cdot d \cdot d \cdot d}$ and ${d^3}$ means ${d \cdot d \cdot d}$. The highest power of ${d}$ that is common to both terms is ${d^3}$. We can factor out ${d^3}$ from both terms. Now that we've identified the common factors, we can factor them out of the entire expression. We're factoring out both 5 and ${d^3}$, so we're essentially factoring out ${5d^3}$. Let's see what happens when we do that:

${5d^3(5d + 4)}$

We've factored out ${5d^3}$ from both terms. To see how we got this, think of it like this:

• When we divide ${25d^4}$ by ${5d^3}$, we get ${5d}$ (because 25 divided by 5 is 5, and ${d^4}$ divided by ${d^3}$ is ${d}$).
• When we divide ${20d^3}$ by ${5d^3}$, we get 4 (because 20 divided by 5 is 4, and ${d^3}$ divided by ${d^3}$ is 1).

So, our simplified expression is ${5d^3(5d + 4)}$. We've successfully factored out the greatest common factor, and the expression inside the parentheses, ${5d + 4}$, cannot be simplified further because ${5d}$ and 4 are unlike terms. There are no more common factors to extract, so we've reached the simplest form of the expression. Remember, identifying the greatest common factor is key to simplifying such expressions efficiently.

Key takeaway: Look for the greatest common factor (GCF) for both the coefficients and the variables. Factoring out the GCF is the most efficient way to simplify these types of expressions.

Conclusion

So, there you have it, guys! We've walked through simplifying three different algebraic expressions. The key to simplifying algebraic expressions is to always look for common factors and like terms. Factoring and combining like terms are your best friends in this process. Remember to take each expression step by step, and you'll become a pro at simplifying in no time! Keep practicing, and you'll be simplifying complex expressions with ease. Happy simplifying!