Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into a common algebraic problem: simplifying expressions. Today, we'll tackle the expression $\frac{b^2}{6 a^2}-\frac{1}{3 a^3}+\frac{3 b}{8 a}$. This isn't as scary as it looks, I promise! We'll break it down step by step to arrive at the simplest form. Our goal is to perform the indicated operations, find a common denominator, combine the terms, and eventually simplify to a single, neat fraction. Get ready to flex those math muscles! This process involves finding a common denominator, adjusting the numerators, and combining like terms. Let's get started. Remember, the key here is to find that least common denominator and then rewrite each fraction with that denominator. This is the cornerstone of adding or subtracting fractions, and it’s the foundation for many more complex algebraic manipulations. Think of it as a puzzle – we’re just rearranging the pieces to make the picture clearer.

Finding the Least Common Denominator (LCD)

Alright, first things first: we need to find the least common denominator (LCD). This is the smallest expression that all the denominators ($6a^2$, $3a^3$, and $8a$) will divide into evenly. Think of it like this: What's the smallest number that 6, 3, and 8 all go into? And how do we account for those 'a's? To find the LCD for the variable parts, consider the highest power of 'a' present in any of the denominators. Our denominators are $6a^2$, $3a^3$, and $8a$. Looking at the coefficients (6, 3, and 8), the least common multiple is 24. Now, let's look at the variable parts: $a^2$, $a^3$, and $a$. The highest power of 'a' is $a^3$. Therefore, the LCD is $24a^3$. This LCD will allow us to combine these fractions into a single term. Before we proceed, let's refresh on what each part of the fraction represents – the numerator, the denominator, and the fraction bar that separates them. Then we will rewrite each fraction with the LCD.

Now we'll rewrite each fraction with the denominator of $24a^3$. This means we will need to change the numerator of each fraction. The concept of the least common denominator is crucial when combining fractions, ensuring that we're adding or subtracting quantities that are directly comparable. It's like comparing apples to apples – you can't simply add the numerators of fractions with different denominators. You have to convert them to a common denominator first, which is what we will do. The process of finding the LCD is also essential in other areas of mathematics, like calculus, where it helps in simplifying more complex expressions.

Rewriting Each Fraction

Okay, now let's rewrite each fraction using the LCD, $24a^3$. We're essentially multiplying each fraction by a form of 1 (a fraction where the numerator and denominator are the same) to get our common denominator. Here's how it breaks down:

1. For the first fraction $\frac{b^2}{6 a^2}$: We need to multiply both the numerator and denominator by $4a$ to get $24a^3$. So, $\frac{b^2}{6 a^2} * \frac{4a}{4a} = \frac{4ab^2}{24a^3}$.

2. For the second fraction $\frac{1}{3 a^3}$: We need to multiply both the numerator and denominator by 8 to get $24a^3$. So, $\frac{1}{3 a^3} * \frac{8}{8} = \frac{8}{24a^3}$.

3. For the third fraction $\frac{3b}{8 a}$: We need to multiply both the numerator and denominator by $3a^2$ to get $24a^3$. So, $\frac{3b}{8 a} * \frac{3a^2}{3a^2} = \frac{9a^2b}{24a^3}$.

See how we've adjusted each fraction so they all have the same denominator? This is the most crucial step, as it allows us to combine them accurately. Remember, what we're doing is the mathematical equivalent of finding a common unit of measure. Before we can add or subtract, all the parts need to be measured in the same units. And now, they are. Each of the numerators is adjusted in accordance with what we had to multiply the denominator by to get to the LCD. This ensures the value of the overall fraction stays the same, even though it looks different. This is the beauty of algebra: the ability to manipulate equations while maintaining their underlying values.

Combining the Fractions

Alright, now that all the fractions share the same denominator ($24a^3$), we can combine them! This is the fun part – we're finally simplifying the expression. Here's how it looks:

rac{4ab^2}{24a^3} - rac{8}{24a^3} + rac{9a^2b}{24a^3} becomes $\frac{4ab^2 - 8 + 9a^2b}{24a^3}$.

We simply combined the numerators, keeping the common denominator. Remember to pay attention to the signs – especially when subtracting. Make sure you carry them through correctly. Be careful with those negative signs, guys! Combining the numerators into a single fraction is the culmination of finding the LCD and rewriting the individual fractions. This results in an expression that is far more manageable and provides a clearer perspective of the relationship between the terms involved. It's like condensing a multi-layered report into a single, comprehensive summary. The combining of fractions is a vital skill that forms the basis for more advanced mathematical operations and concepts.

Now, let's compare our result with the options provided in the question. And that's pretty much it, guys! We have successfully simplified the given expression. This skill is super valuable not just in math but also in various fields where you need to interpret and work with mathematical models. And remember, the more you practice, the easier it gets! This process helps develop a strong foundation in algebra. It emphasizes the importance of understanding the fundamental concepts rather than just memorizing formulas. Remember, practice makes perfect! So, grab some more problems and give them a shot.

Matching with the Options

Now that we've simplified our expression to $\frac{4ab^2 - 8 + 9a^2b}{24a^3}$, let's see which of the provided options matches our result. Looking at the options provided:

A. $\frac{b^2+3 b-1}{11 a^6}$ B. $\frac{b^2-1+3 b}{24 a^3}$ C. $\frac{b^2-1+3 b}{6 a^2-3 a^3+8 a}$ D. $\frac{4 a b^2-8+9 a^2 b}{24 a^3}$

Option D is the perfect match! Our simplified expression is exactly the same as option D. Congratulations, you've successfully simplified the expression and found the correct answer. You can see how the LCD, the rewriting of the fractions, and the combination of the numerators were all important steps in order to arrive at the solution. Also, remember to double-check your work to avoid silly mistakes. Always take the time to compare your solution with the original problem to ensure it makes sense. This final check is very important!

Final Answer

The correct answer is D. $\frac{4 a b^2-8+9 a^2 b}{24 a^3}$

Conclusion

Way to go, everyone! You've successfully navigated the world of simplifying algebraic expressions. We found the LCD, rewrote the fractions, combined them, and ultimately simplified the expression. It's all about breaking down the problem into smaller, manageable steps. Mastering these basics will pave the way for tackling more complex algebraic challenges. Keep practicing, and you'll become a pro in no time! Remember, math is like any other skill: the more you practice, the better you become. I hope you found this guide helpful. Keep learning, keep exploring, and keep the math spirit alive. Until next time, happy calculating!