Simplify Algebraic Expression: ((r - T)³) / (2r + 2t) × (2r) / (r² - Rt)

This article delves into the process of simplifying complex algebraic expressions, specifically focusing on the problem: ((r - t)³) / (2r + 2t) × (2r) / (r² - rt). Simplifying algebraic expressions is a fundamental skill in mathematics, enabling us to manipulate and understand equations more effectively. It involves reducing an expression to its simplest form by canceling out common factors in the numerator and denominator. This process is crucial in various fields, including calculus, physics, and engineering, where complex formulas need to be streamlined for analysis and computation. We'll break down this particular expression step-by-step, explaining each manipulation and the reasoning behind it. Our goal is to transform this multi-term fraction into a concise and manageable form, highlighting techniques such as factoring polynomials and recognizing common algebraic identities. By the end, you'll have a clear understanding of how to approach similar simplification problems, making your mathematical journey smoother and more efficient. Remember, the key to mastering algebraic simplification lies in consistent practice and a solid grasp of factorization techniques. Let's embark on this mathematical adventure together and unravel the mystery behind this expression.

Understanding the Expression

Let's begin by dissecting the given expression: ((r - t)³) / (2r + 2t) × (2r) / (r² - rt). Before we start manipulating, it's essential to recognize the components of this expression. We have a multiplication of two fractions. Each fraction contains a numerator and a denominator, which are themselves polynomials. The first fraction is ((r - t)³) divided by (2r + 2t), and the second fraction is (2r) divided by (r² - rt). Our objective is to combine these into a single, simplified fraction. This involves identifying and canceling out any common factors that appear in both the overall numerator and the overall denominator. This process is akin to simplifying numerical fractions, where we find the greatest common divisor to reduce the fraction to its lowest terms. In algebra, however, we deal with variables and their powers, which adds a layer of complexity but also introduces powerful patterns and techniques for simplification.

Step 1: Factoring the Denominators

The first crucial step in simplifying this algebraic expression is to factorize the denominators as much as possible. This will help us identify common factors that can be canceled out later. Let's look at the first denominator: (2r + 2t). We can see that '2' is a common factor in both terms. Factoring out '2', we get 2(r + t). Now, let's examine the second denominator: (r² - rt). Here, 'r' is a common factor in both terms. Factoring out 'r', we get r(r - t). By performing these factorizations, we've made the structure of the expression more transparent. We can now rewrite the original expression with these factored denominators:

((r - t)³) / 2(r + t) × (2r) / r(r - t)

This rewritten form is a significant step forward. It allows us to see potential cancellations more clearly. Remember, factoring is often the key to unlocking simplification in algebraic expressions. It breaks down complex polynomials into simpler multiplicative components, revealing commonalities that might otherwise be hidden.

Step 2: Expanding and Combining Terms

With the denominators factored, let's rewrite the expression and prepare to combine the numerators and denominators. The expression is now:

((r - t)³) / (2(r + t)) × (2r) / (r(r - t))

To multiply fractions, we multiply the numerators together and the denominators together. So, the combined expression becomes:

( (r - t)³ × 2r ) / ( 2(r + t) × r(r - t) )

Now, let's look for common factors that can be canceled. We have (r - t)³ in the numerator, which is (r - t) × (r - t) × (r - t). In the denominator, we have (r - t). This means we can cancel out one (r - t) term from the numerator with the (r - t) term in the denominator.

Also, we have 2r in the numerator and 2 and r in the denominator. The '2's cancel out, and the 'r's cancel out.

Let's illustrate the cancellation more clearly:

Numerator: (r - t) × (r - t) × (r - t) × 2 × r

Denominator: 2 × (r + t) × r × (r - t)

Canceling 2 from numerator and denominator:

( (r - t)³ × r ) / ( (r + t) × r(r - t) )

Canceling r from numerator and denominator:

( (r - t)³ ) / ( (r + t)(r - t) )

Now, we can cancel one (r - t) from (r - t)³ (which is (r - t) × (r - t) × (r - t)) with the (r - t) in the denominator:

( (r - t)² × (r - t) × (r - t) ) / ( (r + t)(r - t) )

This leaves us with:

( (r - t)² ) / ( r + t )

This simplified form is achieved by systematically canceling out common factors. It's crucial to be meticulous during this stage, ensuring that only identical factors are canceled from both the numerator and the denominator. This step requires careful attention to the powers of the terms and the coefficients.

Step 3: Final Simplification

After performing the cancellations in the previous step, we are left with the expression:

((r - t)²) / (r + t)

Let's double-check if any further simplification is possible. The numerator is (r - t)², which expands to r² - 2rt + t². The denominator is (r + t). There are no common factors between r² - 2rt + t² and r + t. Therefore, the expression is now in its simplest form.

The result is (r - t)² / (r + t).

This final form is considered the most simplified because all possible common factors have been eliminated. The numerator and denominator share no further common terms, making this the most concise representation of the original complex expression. The process of simplification is complete, and we have arrived at a clear and factored answer. This outcome underscores the power of algebraic manipulation and the importance of careful factorization.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying the expression ((r - t)³) / (2r + 2t) × (2r) / (r² - rt) led us to the final answer (r - t)² / (r + t). This journey through algebraic simplification highlights the importance of a systematic approach, beginning with careful factorization of all polynomial components. By identifying common factors in the numerators and denominators, we were able to cancel them out, progressively reducing the complexity of the expression. This technique is not just about arriving at a shorter answer; it's about revealing the underlying structure and relationships within mathematical expressions. Mastery of such simplification skills is fundamental for success in higher mathematics and quantitative fields. It allows for clearer understanding, more efficient problem-solving, and a deeper appreciation for the elegance of algebra. Remember, practice is key. The more you work through various simplification problems, the more intuitive these steps will become. Keep honing your factoring skills and your ability to spot common terms, and you'll find yourself confidently tackling even the most intricate algebraic challenges.

For those interested in further exploring the principles of algebra and simplification, I recommend visiting Khan Academy. Their comprehensive resources offer detailed explanations, practice exercises, and video tutorials that can solidify your understanding of these essential mathematical concepts.