Simplifying Algebraic Fractions Perform The Indicated Operations
In the world of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and make them easier to understand and work with. One common type of problem involves performing indicated operations on fractions with variables, and then simplifying the result. This guide will walk you through a specific example, breaking down each step to ensure clarity and comprehension. So, let's dive in and conquer the art of simplifying algebraic fractions, guys!
The problem we'll be tackling is:
$\frac{3 b}{4 a^2}+\frac{1}{8 a}-\frac{5 b^2}{6 a^3}$
This expression involves three fractions, each with variables in the numerator and denominator. Our goal is to combine these fractions into a single, simplified fraction. To do this, we'll need to find a common denominator, perform the necessary operations, and then simplify the resulting expression. This might sound intimidating, but don't worry! We'll take it one step at a time, making sure you understand each concept along the way. Think of it like building a house – we'll lay the foundation first, and then add the walls and roof.
Finding the Least Common Denominator (LCD)
The least common denominator (LCD) is the smallest expression that all the denominators in our fractions will divide into evenly. Finding the LCD is crucial because it allows us to combine fractions with different denominators. It's like making sure everyone at the party has the same size slice of pizza before we start counting how much pizza is left!
In our problem, the denominators are , , and . To find the LCD, we'll need to consider both the numerical coefficients and the variable terms.
Numerical Coefficients
First, let's look at the numerical coefficients: 4, 8, and 6. We need to find the least common multiple (LCM) of these numbers. The LCM is the smallest number that all three numbers divide into. Here's how we can find it:
- Multiples of 4: 4, 8, 12, 16, 20, 24, ...
- Multiples of 8: 8, 16, 24, 32, ...
- Multiples of 6: 6, 12, 18, 24, 30, ...
The smallest number that appears in all three lists is 24. So, the LCM of 4, 8, and 6 is 24. This means the numerical part of our LCD will be 24.
Variable Terms
Next, we need to consider the variable terms: , , and . To find the LCD for the variables, we take the highest power of each variable that appears in any of the denominators. In this case, the highest power of a is . So, the variable part of our LCD will be .
Combining Numerical and Variable Parts
Now, we combine the numerical and variable parts to get the LCD: 24. This is the expression that all three denominators will divide into evenly. This is our common ground, the shared denominator that will allow us to combine the fractions. It's like having a universal adapter that fits all the different plugs!
Rewriting Fractions with the LCD
Now that we've found the LCD, our next step is to rewrite each fraction with this denominator. This involves multiplying both the numerator and denominator of each fraction by a suitable factor that will result in the LCD in the denominator. Remember, we're not changing the value of the fraction, just its appearance. It's like converting inches to centimeters – the length is the same, but the units are different.
Fraction 1:
To get a denominator of 24, we need to multiply by 6a. So, we multiply both the numerator and denominator by 6a:
$\frac{3 b}{4 a^2} * \frac{6 a}{6 a} = \frac{18 a b}{24 a^3}$
Fraction 2:
To get a denominator of 24, we need to multiply 8a by 3. So, we multiply both the numerator and denominator by 3:
$\frac{1}{8 a} * \frac{3 a^2}{3 a^2} = \frac{3 a^2}{24 a^3}$
Fraction 3:
To get a denominator of 24, we need to multiply 6 by 4. So, we multiply both the numerator and denominator by 4:
$\frac{5 b^2}{6 a^3} * \frac{4}{4} = \frac{20 b^2}{24 a^3}$
Now we have all three fractions with the same denominator: , , and . This is a crucial step because it allows us to combine the fractions. It's like having all the ingredients for a cake – now we can finally mix them together!
Combining the Fractions
Now that all the fractions have the same denominator, we can combine them by adding or subtracting the numerators. The denominator remains the same. This is like adding slices of the same pizza – we just add the numerators (number of slices) and keep the denominator (size of each slice) the same.
Our expression now looks like this:
$\frac{18 a b}{24 a^3} + \frac{3 a^2}{24 a^3} - \frac{20 b^2}{24 a^3}$
Combining the numerators, we get:
$\frac{18 a b + 3 a^2 - 20 b^2}{24 a^3}$
This is a single fraction, but we're not done yet! We need to simplify it as much as possible. It's like baking the cake – we've mixed the ingredients, but we still need to put it in the oven and let it bake.
Simplifying the Result
Simplifying a fraction means reducing it to its simplest form. This involves looking for common factors in the numerator and denominator and canceling them out. It's like cutting the cake into equal slices – we want to make sure everyone gets a fair share.
Factoring (if possible)
First, we try to factor the numerator and denominator. Factoring means breaking down an expression into its constituent parts (factors). This can help us identify common factors that we can cancel out.
In our case, the numerator is . This expression doesn't have any obvious common factors, and it doesn't fit any common factoring patterns (like difference of squares or perfect square trinomial). So, we can't factor the numerator easily. This is like trying to cut a very dense cake – sometimes it's just too difficult!
The denominator is 24. We can factor this as . This will help us identify any common factors with the numerator.
Canceling Common Factors
Since we couldn't factor the numerator, we'll look for common factors between the entire numerator and the denominator. In this case, there are no common factors between the numerator () and the denominator (24). This means we can't simplify the fraction any further. It's like taking the cake out of the oven – it's fully baked, and we can't change its shape anymore.
Final Answer
Since we couldn't simplify the fraction any further, our final answer is:
$\frac{18 a b + 3 a^2 - 20 b^2}{24 a^3}$
This is the simplified form of the original expression. We've successfully performed the indicated operations and simplified the result! Give yourself a pat on the back, guys! You've conquered a challenging algebraic problem.
Key Takeaways
Let's recap the key steps we took to solve this problem:
- Find the Least Common Denominator (LCD): Identify the smallest expression that all denominators divide into evenly.
- Rewrite Fractions with the LCD: Multiply the numerator and denominator of each fraction by a suitable factor to get the LCD in the denominator.
- Combine the Fractions: Add or subtract the numerators, keeping the denominator the same.
- Simplify the Result: Factor the numerator and denominator (if possible) and cancel out any common factors.
By following these steps, you can simplify a wide variety of algebraic fractions. Remember, practice makes perfect! The more you work with these types of problems, the more comfortable you'll become with the process.
Practice Problems
To solidify your understanding, try simplifying the following expressions:
Work through these problems step-by-step, and don't be afraid to make mistakes! Mistakes are a valuable learning opportunity. The important thing is to understand the process and keep practicing.
Conclusion
Simplifying algebraic fractions can seem daunting at first, but with a systematic approach and a bit of practice, it becomes a manageable skill. By understanding the concepts of LCD, rewriting fractions, combining, and simplifying, you can confidently tackle these types of problems. So, keep practicing, stay curious, and never stop learning! You've got this, guys!