How To Find The Quotient And Simplify (b-7)/b^2 ÷ (b^2-49)/(6b)

Aug 2, 2025 by ADMIN 64 views

Finding the Quotient and Simplifying: A Step-by-Step Guide

Hey guys! Today, we're diving into a common math problem: finding the quotient and simplifying expressions. Specifically, we'll be tackling a division problem involving fractions with variables. So, grab your pencils, and let's get started!

Understanding the Problem: Dividing Rational Expressions

Before we jump into the nitty-gritty, let's make sure we're all on the same page. We're dealing with rational expressions, which are basically fractions where the numerator and denominator are polynomials. Our goal is to divide these expressions and then simplify the result as much as possible. This involves a few key steps, and we'll break each one down.

Our specific problem is:

$\frac{b-7}{b^2} \div \frac{b^2-49}{6 b}$

Don't worry if it looks a bit intimidating at first! We're going to take it one step at a time. The most important thing to remember when dividing fractions is that we actually multiply by the reciprocal of the second fraction. Think of it like this: dividing by something is the same as multiplying by its inverse.

This reciprocal trick is the cornerstone of how we handle this problem. It transforms our division problem into a multiplication problem, which we can then tackle using our familiar rules for multiplying fractions. We'll also need to factor polynomials to simplify things later on, so keep that in mind as we proceed.

So, before we dive into the actual steps, let's quickly recap the key concepts. We're dealing with rational expressions, which are fractions containing polynomials. We need to find the quotient, which means we're dividing. And remember, dividing fractions is the same as multiplying by the reciprocal.

Now, let's roll up our sleeves and solve this thing!

Step 1: Rewrite Division as Multiplication

Okay, first things first, let's turn that division problem into a multiplication problem. As we discussed, we do this by taking the reciprocal of the second fraction and multiplying. So, the reciprocal of $\frac{b^2-49}{6 b}$ is $\frac{6b}{b^2-49}$ .

Now, we can rewrite our problem as:

$\frac{b-7}{b^2} \times \frac{6b}{b^2-49}$

See? It already looks a bit friendlier, right? We've replaced the division sign with a multiplication sign, and we're ready to move on to the next step. This is a crucial step because it sets us up for simplifying the expression later. Multiplying fractions is generally easier to handle than dividing them, especially when dealing with algebraic expressions.

This might seem like a small step, but it's a powerful one. By changing the operation, we open the door to using the rules of fraction multiplication, which are much more straightforward in this context. This reciprocal trick is a fundamental concept in working with fractions, so make sure you've got it down!

So, to recap this step, we took our original division problem and transformed it into a multiplication problem by using the reciprocal of the second fraction. We now have a multiplication problem that we can tackle in the next step. We're making progress, guys!

Step 2: Factor the Expressions

Now comes the fun part: factoring! Factoring is the process of breaking down a polynomial into its constituent factors. This is crucial for simplifying rational expressions because it allows us to identify common factors in the numerator and denominator that we can cancel out.

Looking at our expression, $\frac{b-7}{b^2} \times \frac{6b}{b^2-49}$ , we see that the term $b^2 - 49$ can be factored. This is a classic example of the difference of squares pattern, which states that $a^2 - b^2 = (a + b)(a - b)$ .

In our case, $b^2 - 49$ is the same as $b^2 - 7^2$ , so we can factor it as $(b + 7)(b - 7)$ .

Our expression now becomes:

$\frac{b-7}{b^2} \times \frac{6b}{(b + 7)(b - 7)}$

Notice how factoring has revealed a common factor of $(b - 7)$ in both the numerator and denominator. This is exactly what we were hoping for! Factoring is like unlocking a secret code that allows us to simplify the expression.

Also, we can rewrite $b^2$ as $b \cdot b$ to make the simplification in the next step even clearer. Our expression now looks like this:

$\frac{b-7}{b \cdot b} \times \frac{6b}{(b + 7)(b - 7)}$

Factoring is a fundamental skill in algebra, and it's essential for simplifying rational expressions. By factoring, we can identify common factors that can be canceled, making the expression much simpler to work with. So, make sure you're comfortable with factoring techniques, like the difference of squares, before moving on!

Step 3: Simplify by Canceling Common Factors

Alright, we've set the stage, and now it's time for the grand finale of simplification! Remember those common factors we identified in the previous step? Now we get to cancel them out. This is where all our hard work pays off, and the expression starts to look much cleaner.

Looking at our expression:

$\frac{b-7}{b \cdot b} \times \frac{6b}{(b + 7)(b - 7)}$

We can see that $(b - 7)$ appears in both the numerator and the denominator, so we can cancel them out. Also, we have a factor of $b$ in both the numerator and the denominator, so we can cancel one of those as well.

After canceling, we're left with:

$\frac{1}{b} \times \frac{6}{(b + 7)}$

See how much simpler that looks? By canceling common factors, we've reduced the expression to its most basic form. This is the essence of simplification – making things as clear and concise as possible.

It's important to remember that we can only cancel factors that are multiplied, not terms that are added or subtracted. This is why factoring is so important; it allows us to identify those multiplicative factors that we can cancel.

So, we've successfully canceled out the common factors and simplified our expression. We're almost there! Now, all that's left is to multiply the remaining fractions together.

Step 4: Multiply the Remaining Fractions

We're in the home stretch now! We've canceled out the common factors, and we're left with two simple fractions to multiply. Remember, to multiply fractions, we simply multiply the numerators and multiply the denominators.

Our simplified expression is:

$\frac{1}{b} \times \frac{6}{(b + 7)}$

Multiplying the numerators, we get $1 \times 6 = 6$ .

Multiplying the denominators, we get $b \times (b + 7) = b(b + 7)$ .

So, our final simplified expression is:

$\frac{6}{b(b + 7)}$

And that's it! We've found the quotient and simplified it as much as possible. We've successfully navigated the division of rational expressions, factored polynomials, canceled common factors, and multiplied the remaining fractions. Give yourselves a pat on the back, guys!

Final Answer

Therefore, the quotient of $\frac{b-7}{b^2} \div \frac{b^2-49}{6 b}$ , simplified, is $\frac{6}{b(b + 7)}$ .

Key Takeaways

Let's quickly recap the key steps we took to solve this problem:

Rewrite division as multiplication: Multiply by the reciprocal of the second fraction.
Factor the expressions: Look for patterns like the difference of squares.
Simplify by canceling common factors: Identify and cancel factors that appear in both the numerator and denominator.
Multiply the remaining fractions: Multiply the numerators and multiply the denominators.

By following these steps, you can confidently tackle division problems involving rational expressions. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time!

Practice Problems

Want to test your skills? Try these practice problems:

$\frac{x+2}{x^2} \div \frac{x^2-4}{3x}$
$\frac{y-5}{y^2-25} \div \frac{y}{2y+10}$

Good luck, and happy simplifying!