Equivalent Expression: Solving A Rational Expression

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Hey guys! Let's dive into a fun math problem today that involves simplifying a complex rational expression. We're going to break down the steps to figure out which expression is equivalent to the given one. So, grab your pencils, and let's get started!

Understanding the Problem

Our main question is: what expression is equivalent to the following?

m+3m2−16m2−9m+4\frac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}}

Before we jump into solving it, let’s take a quick look at the options we have:

A. 1(m+4)(m+3)\frac{1}{(m+4)(m+3)} B. 1(m−4)(m−3)\frac{1}{(m-4)(m-3)} C. m−4m−3\frac{m-4}{m-3} D. m+3m+4\frac{m+3}{m+4}

To find the correct answer, we'll need to simplify the original expression step by step. This involves dealing with fractions within fractions, which might seem a bit tricky, but don't worry, we'll tackle it together!

Step-by-Step Solution

Step 1: Rewrite the Complex Fraction

The first thing we need to do is rewrite the complex fraction as a division problem. Remember, a fraction bar means division, so we can rewrite the expression as:

m+3m2−16÷m2−9m+4\frac{m+3}{m^2-16} \div \frac{m^2-9}{m+4}

This makes it a bit easier to see what we need to do next.

Step 2: Change Division to Multiplication

When we divide fractions, we actually multiply by the reciprocal of the second fraction. So, we flip the second fraction and change the division sign to a multiplication sign:

m+3m2−16×m+4m2−9\frac{m+3}{m^2-16} \times \frac{m+4}{m^2-9}

Now our expression looks like a regular multiplication problem involving fractions.

Step 3: Factor the Expressions

Next, we need to factor the expressions in the numerators and denominators. Factoring helps us simplify by canceling out common terms. Let's break down each part:

  • m2−16m^2 - 16 is a difference of squares, which factors into (m+4)(m−4)(m+4)(m-4).
  • m2−9m^2 - 9 is also a difference of squares, factoring into (m+3)(m−3)(m+3)(m-3).

So, our expression now looks like this:

m+3(m+4)(m−4)×m+4(m+3)(m−3)\frac{m+3}{(m+4)(m-4)} \times \frac{m+4}{(m+3)(m-3)}

Step 4: Cancel Out Common Factors

Now comes the fun part – canceling out the common factors! We have (m+3)(m+3) in both the numerator and denominator, and we also have (m+4)(m+4) in both. Let's cancel them out:

(m+3)(m+4)(m−4)×(m+4)(m+3)(m−3)\frac{\cancel{(m+3)}}{\cancel{(m+4)}(m-4)} \times \frac{\cancel{(m+4)}}{\cancel{(m+3)}(m-3)}

After canceling, we are left with:

1(m−4)(m−3)\frac{1}{(m-4)(m-3)}

Step 5: Identify the Equivalent Expression

Looking back at our options, we can see that the simplified expression matches option B:

B. 1(m−4)(m−3)\frac{1}{(m-4)(m-3)}

So, the equivalent expression is 1(m−4)(m−3)\frac{1}{(m-4)(m-3)}.

Why This Problem Matters

Rational expressions might seem abstract, but they're super useful in many areas of math and science. They pop up in calculus, physics, and engineering, especially when you're dealing with rates, proportions, and complex relationships. Mastering how to simplify these expressions helps you tackle tougher problems later on. Plus, it sharpens your algebra skills, making you a more confident problem-solver overall!

Common Mistakes to Avoid

When working with rational expressions, it's easy to slip up if you're not careful. Here are a few common mistakes to watch out for:

  • Forgetting to flip when dividing: Remember, dividing fractions means multiplying by the reciprocal. If you forget to flip the second fraction, you'll end up with the wrong answer.
  • Not factoring correctly: Factoring is key to simplifying these expressions. Make sure you know your factoring rules, especially the difference of squares.
  • Canceling terms incorrectly: You can only cancel factors, not terms. For example, you can't cancel the mm in (m−4)(m-4) because it's part of a term, not a factor.
  • Ignoring excluded values: Remember that the denominator of a fraction cannot be zero. So, you need to consider values of mm that would make the denominator zero and exclude them from your solution.

Practice Problems

To really nail this concept, let's try a few more problems. Practice makes perfect, and the more you work with these expressions, the easier they'll become.

Problem 1

Simplify the expression:

x+2x2−9x2−4x−3\frac{\frac{x+2}{x^2-9}}{\frac{x^2-4}{x-3}}

Solution

First, rewrite the expression as a division problem:

x+2x2−9÷x2−4x−3\frac{x+2}{x^2-9} \div \frac{x^2-4}{x-3}

Next, change division to multiplication and flip the second fraction:

x+2x2−9×x−3x2−4\frac{x+2}{x^2-9} \times \frac{x-3}{x^2-4}

Now, factor all the expressions:

x+2(x+3)(x−3)×x−3(x+2)(x−2)\frac{x+2}{(x+3)(x-3)} \times \frac{x-3}{(x+2)(x-2)}

Cancel out common factors:

(x+2)(x+3)(x−3)×(x−3)(x+2)(x−2)\frac{\cancel{(x+2)}}{(x+3)\cancel{(x-3)}} \times \frac{\cancel{(x-3)}}{\cancel{(x+2)}(x-2)}

Simplify:

1(x+3)(x−2)\frac{1}{(x+3)(x-2)}

Problem 2

Simplify the expression:

a−5a2−1a2−25a+1\frac{\frac{a-5}{a^2-1}}{\frac{a^2-25}{a+1}}

Solution

Rewrite as a division problem:

a−5a2−1÷a2−25a+1\frac{a-5}{a^2-1} \div \frac{a^2-25}{a+1}

Change to multiplication and flip:

a−5a2−1×a+1a2−25\frac{a-5}{a^2-1} \times \frac{a+1}{a^2-25}

Factor:

a−5(a+1)(a−1)×a+1(a+5)(a−5)\frac{a-5}{(a+1)(a-1)} \times \frac{a+1}{(a+5)(a-5)}

Cancel common factors:

(a−5)(a+1)(a−1)×(a+1)(a+5)(a−5)\frac{\cancel{(a-5)}}{\cancel{(a+1)}(a-1)} \times \frac{\cancel{(a+1)}}{(a+5)\cancel{(a-5)}}

Simplify:

1(a−1)(a+5)\frac{1}{(a-1)(a+5)}

Tips for Mastering Rational Expressions

  • Review factoring: Make sure you're comfortable with factoring different types of expressions, like the difference of squares, perfect square trinomials, and simple quadratics.
  • Practice regularly: The more you practice, the more comfortable you'll become with the steps involved in simplifying rational expressions.
  • Write neatly: It's easy to make mistakes if your work is messy. Keep your steps organized and write clearly.
  • Check your work: After simplifying an expression, plug in a value for the variable to see if your simplified expression matches the original. This can help you catch errors.
  • Understand the restrictions: Always remember to consider the values that would make the denominator zero and exclude them from your solution.

Conclusion

Simplifying rational expressions is a fundamental skill in algebra. By breaking down the problem into steps—rewriting as division, changing to multiplication, factoring, canceling, and simplifying—you can tackle even complex expressions with confidence. Remember to watch out for common mistakes and practice regularly. Keep up the great work, and you'll master these expressions in no time!

So, to recap, the expression equivalent to m+3m2−16m2−9m+4\frac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}} is indeed B. 1(m−4)(m−3)\frac{1}{(m-4)(m-3)}. You guys nailed it! Keep practicing, and you'll become a pro at simplifying rational expressions.