Multiplying Rational Expressions Simplifying Products

by ADMIN 54 views

Multiply rational expressions is a fundamental operation in algebra, often encountered in various mathematical contexts. The given problem, $\frac{30 a3}{a2-16} \cdot \frac{a^2+4 a}{24 a^3}$, exemplifies this operation, requiring us to find the product of two rational expressions and simplify the result to its simplest form. This process involves several key steps, including factoring, identifying common factors, and canceling them out to arrive at the most reduced expression. Understanding how to multiply rational expressions and simplify the results is crucial for solving more complex algebraic problems and gaining a deeper understanding of rational functions.

To effectively tackle this problem, we need to break down each component of the expression. The first rational expression is $\frac{30 a3}{a2-16}$, and the second is $\frac{a^2+4 a}{24 a^3}$. The task is to multiply these two rational expressions together and then simplify the resulting expression. Simplification is a critical step because it ensures that the final answer is presented in the most concise and understandable form. This involves identifying and canceling out any common factors between the numerator and the denominator. Mastery of this process is essential for anyone studying algebra, as it lays the groundwork for more advanced topics such as solving rational equations and working with rational functions.

When we multiply rational expressions, we essentially multiply the numerators together and the denominators together. Before doing this directly, it is often beneficial to factor the expressions in the numerators and denominators. Factoring allows us to identify common factors more easily, which can then be canceled out. In our problem, we have $a^2 - 16$ in the denominator of the first fraction, which is a difference of squares and can be factored as $(a - 4)(a + 4)$. In the numerator of the second fraction, we have $a^2 + 4a$, which can be factored by taking out the common factor $a$, resulting in $a(a + 4)$. Factoring these expressions is a key step in simplifying the overall product. By expressing the polynomials in their factored forms, we can readily see the common factors that can be canceled out, leading to a simplified final expression. This technique is widely applicable in algebra and is a fundamental skill for simplifying rational expressions.

Step-by-step Solution

Let's walk through the step-by-step solution to multiply the rational expressions and simplify the product:

  1. Write the given expression: We start with the expression:

    30a3a2−16⋅a2+4a24a3\frac{30 a^3}{a^2-16} \cdot \frac{a^2+4 a}{24 a^3}

  2. Factor the expressions: Factor the denominator $a^2 - 16$ and the numerator $a^2 + 4a$. We recognize that $a^2 - 16$ is a difference of squares, which factors into $(a - 4)(a + 4)$. The expression $a^2 + 4a$ can be factored by taking out the common factor $a$, giving us $a(a + 4)$.

    30a3(a−4)(a+4)⋅a(a+4)24a3\frac{30 a^3}{(a-4)(a+4)} \cdot \frac{a(a+4)}{24 a^3}

  3. Multiply the numerators and denominators: Multiply the numerators together and the denominators together:

    30a3⋅a(a+4)(a−4)(a+4)⋅24a3\frac{30 a^3 \cdot a(a+4)}{(a-4)(a+4) \cdot 24 a^3}

  4. Simplify the expression: Now, we simplify the expression by canceling out common factors. We can cancel out $a^3$ from both the numerator and the denominator. We can also cancel out the factor $(a + 4)$ from both the numerator and the denominator. Additionally, we can simplify the numerical coefficients by dividing both 30 and 24 by their greatest common divisor, which is 6. This gives us 5 in the numerator and 4 in the denominator.

    30a4(a+4)24a3(a−4)(a+4)=5a4(a+4)4a3(a−4)(a+4)\frac{30 a^4 (a+4)}{24 a^3 (a-4)(a+4)} = \frac{5 a^4 (a+4)}{4 a^3 (a-4)(a+4)}

    5a3a(a+4)4a3(a−4)(a+4)=5a4(a−4)\frac{5 \cancel{a^3} a \cancel{(a+4)}}{4 \cancel{a^3} (a-4) \cancel{(a+4)}} = \frac{5a}{4(a-4)}

  5. Final Simplification: Distribute the 4 in the denominator:

    5a4(a−4)=5a4a−16\frac{5a}{4(a-4)} = \frac{5a}{4a-16}

Thus, the product in simplest form is $\frac{5a}{4a-16}$.

Evaluating the Answer Choices

Now, let's evaluate the given answer choices:

A. $ rac{5}{a-4}$ B. $ rac{5 a^2}{4 a^2-16 a}$ C. $ rac{5 a}{-16}$ D. $ rac{5 a}{4 a-16}$

Comparing our simplified expression $\frac{5a}{4a-16}$ with the answer choices, we can see that option D matches our result. Therefore, the correct answer is D.

Common Mistakes and How to Avoid Them

When multiplying and simplifying rational expressions, several common mistakes can occur. Understanding these pitfalls and how to avoid them is crucial for achieving accurate results. One frequent error is failing to factor the expressions completely before canceling out common factors. For instance, overlooking the difference of squares pattern or not factoring out the greatest common factor can lead to incorrect simplifications. Another common mistake is incorrectly canceling terms instead of factors. Only factors that are multiplied can be canceled; terms that are added or subtracted cannot be canceled. For example, in the expression $ rac{a + 4}{a}$, you cannot cancel the $a$'s because the 4 is added to $a$ in the numerator.

To avoid these mistakes, always begin by factoring the numerators and denominators completely. This makes it easier to identify common factors. Double-check your factoring to ensure accuracy. Then, carefully cancel out the common factors, making sure you are only canceling factors, not terms. A systematic approach, such as writing out each step clearly, can also help prevent errors. Finally, always double-check your final answer to ensure that the expression is indeed in its simplest form. By being meticulous and understanding the rules of algebraic manipulation, you can minimize the chances of making mistakes and confidently multiply and simplify rational expressions.

Real-World Applications of Rational Expressions

Rational expressions are not just abstract algebraic concepts; they have numerous real-world applications in various fields. Understanding how to multiply and simplify rational expressions can be incredibly useful in practical scenarios. One significant application is in physics, where rational expressions are used to describe relationships between physical quantities such as velocity, time, and distance. For example, the formula for average speed, which is distance divided by time, is a rational expression. Similarly, in engineering, rational expressions are used to model and analyze electrical circuits, fluid dynamics, and structural mechanics.

In economics and business, rational expressions can be used to model cost functions, revenue functions, and profit functions. For instance, the average cost of producing a certain number of items can be represented as a rational expression, where the total cost is divided by the number of items produced. In computer science, rational expressions are used in algorithm analysis and complexity theory. The efficiency of an algorithm can often be described using rational expressions that relate the input size to the time or space complexity. These are just a few examples of how rational expressions play a crucial role in various disciplines, highlighting the importance of mastering the skills to multiply and simplify rational expressions for real-world problem-solving.

Conclusion

In conclusion, multiplying rational expressions and simplifying the product is a fundamental skill in algebra with wide-ranging applications. By following a systematic approach, which includes factoring, identifying common factors, canceling, and simplifying, we can accurately find the product in its simplest form. The given problem, $\frac{30 a3}{a2-16} \cdot \frac{a^2+4 a}{24 a^3}$, serves as a comprehensive example of this process. Through careful step-by-step simplification, we determined that the correct answer is $\frac{5 a}{4 a-16}$, which corresponds to option D. Avoiding common mistakes, such as failing to factor completely or incorrectly canceling terms, is crucial for success. Understanding the real-world applications of rational expressions further underscores the importance of mastering this algebraic skill. Whether in physics, engineering, economics, or computer science, the ability to multiply and simplify rational expressions is a valuable tool for problem-solving and analysis.