Simplifying Algebraic Expressions: A Step-by-Step Guide

Oct 30, 2025 by Admin 56 views

Hey there, math enthusiasts! Today, we're diving into the world of algebraic expressions, specifically focusing on how to simplify expressions like $\left(\frac{4}{5} a\right)\left(\frac{8}{9} a\right)$ . Don't worry, it's not as scary as it looks! We'll break it down step-by-step, making sure you understand the process and can confidently tackle similar problems in the future. So, grab your pencils and let's get started!

Understanding the Basics: Algebraic Expressions

Before we jump into the simplification, let's make sure we're all on the same page about what an algebraic expression actually is. In simple terms, an algebraic expression is a combination of numbers, variables (represented by letters like 'a', 'x', or 'y'), and mathematical operations (like addition, subtraction, multiplication, and division). Think of it as a mathematical phrase. For example, $2x + 3$ , $5y - 7$ , and, of course, $\left(\frac{4}{5} a\right)\left(\frac{8}{9} a\right)$ are all algebraic expressions.

The key to understanding and working with algebraic expressions lies in the ability to identify the different components and how they interact. We have constants which are numbers, variables represented by letters, and coefficients, which are the numbers multiplying the variables. The operations tell us what we should do with these components.

Now, simplifying an algebraic expression means rewriting it in a more concise or manageable form. This often involves combining like terms, which are terms that have the same variable raised to the same power. For instance, in the expression $3x + 2x + 5$ , the terms $3x$ and $2x$ are like terms and can be combined to give $5x$ . The constant term, 5, remains as is because it doesn't have a variable attached. The goal is to make the expression as simple and easy to understand as possible. It’s like tidying up a messy room – you’re not changing what's there, just organizing it in a more efficient way. In the context of our specific problem, simplifying $\left(\frac{4}{5} a\right)\left(\frac{8}{9} a\right)$ involves performing the multiplication and rewriting the expression in a simplified format. This process will make the expression easier to work with, to evaluate for a specific value of ‘a’, or to integrate into a larger equation.

Breaking Down the Problem: $\left(\frac{4}{5} a\right)\left(\frac{8}{9} a\right)$

Okay, guys, let's take a closer look at our problem: $\left(\frac{4}{5} a\right)\left(\frac{8}{9} a\right)$ . The expression involves the multiplication of two terms, each composed of a fraction and a variable. Our objective is to simplify this expression by performing the multiplication. Remember, the multiplication of algebraic terms is based on two key principles: the multiplication of coefficients (the numbers) and the multiplication of variables.

Let's go through the steps of simplification so that it is simple to follow. First, we multiply the coefficients: $\frac{4}{5} \times \frac{8}{9}$ . Remember how to multiply fractions? You simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, $4 \times 8 = 32$ and $5 \times 9 = 45$ . This gives us a new coefficient of $\frac{32}{45}$ .

Next, we multiply the variables. In this case, we have $a \times a$ . When you multiply a variable by itself, you raise it to the power of 2. So, $a \times a = a^2$ . Combining the results from the multiplication of the coefficients and the variables, we get the simplified expression. This is because the $a$ has a power of 1 and when multiplying the same variables we simply sum the powers.

Step-by-Step Simplification

Alright, let's get down to the nitty-gritty and work through the simplification step-by-step. This is where the rubber meets the road, so pay close attention!

Multiply the coefficients: We start by multiplying the fractional coefficients: $\frac{4}{5} \times \frac{8}{9}$ . To do this, we multiply the numerators together and the denominators together. So, we have: $4 \times 8 = 32$ and $5 \times 9 = 45$ . This gives us a new coefficient of $\frac{32}{45}$ .
Multiply the variables: Next, we need to multiply the variables together: $a \times a$ . When you multiply a variable by itself, you raise it to the power of 2. Therefore, $a \times a = a^2$ .
Combine the results: Now, let's combine the results from steps 1 and 2. We have the coefficient $\frac{32}{45}$ and the variable part $a^2$ . Putting them together, we get the simplified expression $\frac{32}{45} a^2$ .

So, the simplified form of $\left(\frac{4}{5} a\right)\left(\frac{8}{9} a\right)$ is $\frac{32}{45} a^2$ . And there you have it, folks! We've successfully simplified the expression. It's not so hard once you break it down into smaller, manageable steps, right? The key is to remember the rules of multiplying fractions and variables. This skill is critical for any future algebraic manipulation.

Selecting the Correct Answer

Now, let's look at the multiple-choice options and select the correct answer. The options are:

A. $\frac{64}{45} a^2$ B. $\frac{64}{45}$ C. $\frac{32}{45} a$ D. $\frac{32}{45} a^2$

Based on our calculations, the correct answer is D. $\frac{32}{45} a^2$ . This is because when we multiplied $\frac{4}{5} a$ by $\frac{8}{9} a$ , we obtained $\frac{32}{45} a^2$ . Remember, always double-check your calculations and make sure your answer matches the simplified expression you derived. If you made a mistake at any point, retrace your steps and find the error. Sometimes a simple calculation can be overlooked, so carefulness is essential.

Tips and Tricks for Simplifying Algebraic Expressions

Here are some handy tips and tricks to help you become a simplification superstar:

Practice Makes Perfect: The more you practice, the better you'll become at simplifying expressions. Work through various examples, and don't be afraid to make mistakes – that's how you learn!
Break It Down: If an expression looks complicated, break it down into smaller, more manageable parts. This will make the process less overwhelming.
Double-Check Your Work: Always double-check your calculations, especially when dealing with fractions and exponents. It's easy to make a small error that can change the entire answer.
Know Your Rules: Make sure you understand the basic rules of algebra, such as the order of operations (PEMDAS/BODMAS), how to multiply fractions, and how to work with exponents and variables.
Use Visual Aids: If you're a visual learner, try using diagrams or color-coding to help you understand the different parts of an expression.

Conclusion

Simplifying algebraic expressions might seem daunting at first, but with a clear understanding of the rules and some practice, you'll be able to solve these problems. We've gone over the essential steps for simplifying expressions. Remember to take it slow, break down the problem into smaller parts, and always double-check your work. You've got this!

Keep practicing, and you'll become a pro at simplifying algebraic expressions in no time! Keep experimenting with different expressions to test your skills and solidify your understanding. Now that you are equipped with the knowledge and tools, go forth and conquer those algebraic expressions!