Evaluating The Algebraic Expression (1.15^3 - 0.15^3) / (1.35^2 - 0.35^2) * 1.7 / (1.15^2 + 1.15 * 0.15 + 0.15 * 0.15)
Introduction
In this article, we will delve into the intricate world of algebraic expressions and explore the value of a specific expression. Our focus will be on unraveling the expression:
(1.15^3 - 0.15^3) / (1.35^2 - 0.35^2) * 1.7 / (1.15^2 + 1.15 * 0.15 + 0.15 * 0.15).
This expression may appear daunting at first glance, but with a strategic approach and the application of algebraic identities, we can simplify it and determine its numerical value. The problem falls under the discussion category of mathematics, specifically focusing on algebraic manipulations and arithmetic calculations. This problem is a great example of how understanding fundamental mathematical concepts and applying them skillfully can lead to elegant solutions. We will explore how to use the difference of cubes factorization and the difference of squares factorization to simplify the expression. Furthermore, we will emphasize the importance of order of operations (PEMDAS/BODMAS) to ensure accurate calculations. By breaking down each step, we aim to provide a clear and concise explanation that will help readers not only understand the solution but also appreciate the underlying mathematical principles. Whether you're a student looking to improve your algebra skills or simply someone who enjoys mathematical challenges, this article will offer valuable insights into the world of algebraic problem-solving.
Breaking Down the Expression
To effectively tackle this expression, we will first break it down into smaller, more manageable parts. Our primary strategy involves applying algebraic identities to simplify both the numerator and the denominator of the fractions involved. This simplification process is crucial for making the calculation more straightforward and less prone to errors. We will begin by focusing on the first fraction, which involves the difference of cubes in the numerator and the difference of squares in the denominator. These are classic algebraic patterns that, when recognized and applied correctly, can significantly simplify complex expressions. The difference of cubes identity, a^3 - b^3 = (a - b)(a^2 + ab + b^2), will be our key tool for simplifying the numerator. Similarly, the difference of squares identity, a^2 - b^2 = (a + b)(a - b), will be instrumental in simplifying the denominator. By applying these identities, we transform the expression into a form where we can identify common factors and potentially cancel them out. This is a common technique in algebraic simplification and is essential for solving problems efficiently. Furthermore, we will pay close attention to the numerical values involved, ensuring that our calculations are accurate and precise. This step-by-step approach not only simplifies the expression but also reduces the risk of making mistakes along the way. By breaking down the problem into smaller parts, we can focus on each element individually and then combine the results to arrive at the final answer. This methodical approach is a hallmark of effective problem-solving in mathematics.
Applying Algebraic Identities
Simplifying the Numerator
Let's begin by focusing on the numerator of the first fraction, which is 1.15^3 - 0.15^3. This expression perfectly fits the form of the difference of cubes identity: a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here, a = 1.15 and b = 0.15. Applying the identity, we get:
- 15^3 - 0.15^3 = (1.15 - 0.15)(1.15^2 + 1.15 * 0.15 + 0.15^2)
Simplifying the first term (1.15 - 0.15), we get 1. Therefore, the expression becomes:
1 * (1.15^2 + 1.15 * 0.15 + 0.15^2) = (1.15^2 + 1.15 * 0.15 + 0.15^2)
This transformation is a crucial step in simplifying the overall expression. By applying the difference of cubes identity, we have converted a complex subtraction of cubes into a more manageable form involving squares and products. This new form is easier to work with and allows us to further simplify the expression in subsequent steps. The ability to recognize and apply algebraic identities like the difference of cubes is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. This step not only simplifies the numerator but also reveals a pattern that will be helpful later when we address the denominator. The careful application of algebraic identities is a powerful tool in simplifying complex mathematical expressions, making them easier to analyze and solve.
Simplifying the Denominator
Now, let's turn our attention to the denominator of the first fraction, which is 1.35^2 - 0.35^2. This expression is in the form of the difference of squares, a^2 - b^2, where a = 1.35 and b = 0.35. We can apply the difference of squares identity, which states that a^2 - b^2 = (a + b)(a - b). Substituting the values of a and b, we get:
- 35^2 - 0.35^2 = (1.35 + 0.35)(1.35 - 0.35)
Now, we simplify the terms inside the parentheses:
- 35 + 0.35 = 1.7
- 35 - 0.35 = 1
Therefore, the denominator simplifies to:
(1.7)(1) = 1.7
This simplification is another key step in solving the problem. By applying the difference of squares identity, we have transformed the denominator into a simple product, which is much easier to handle. The ability to recognize and apply such identities is crucial in algebraic manipulations. This step demonstrates the power of algebraic identities in simplifying complex expressions and making them more manageable. The result we obtained for the denominator, 1.7, is particularly interesting because it appears as a factor in another part of the original expression. This suggests that there may be further simplifications possible when we combine the simplified numerator and denominator. The application of the difference of squares identity not only simplifies the denominator but also sets the stage for further simplification by revealing common factors within the overall expression.
Combining and Simplifying
Now that we have simplified both the numerator and the denominator of the first fraction, we can combine them and proceed with further simplification. Recall that the original expression is:
(1.15^3 - 0.15^3) / (1.35^2 - 0.35^2) * 1.7 / (1.15^2 + 1.15 * 0.15 + 0.15^2)
We found that:
- The numerator (1.15^3 - 0.15^3) simplifies to (1.15^2 + 1.15 * 0.15 + 0.15^2)
- The denominator (1.35^2 - 0.35^2) simplifies to 1.7
Substituting these simplified expressions back into the original expression, we get:
[(1.15^2 + 1.15 * 0.15 + 0.15^2) / 1.7] * [1.7 / (1.15^2 + 1.15 * 0.15 + 0.15^2)]
Notice that the term (1.15^2 + 1.15 * 0.15 + 0.15^2) appears in both the numerator of the first fraction and the denominator of the second fraction. Similarly, the term 1.7 appears in the denominator of the first fraction and the numerator of the second fraction. This allows us to cancel out these common terms:
[(1.15^2 + 1.15 * 0.15 + 0.15^2) / 1.7] * [1.7 / (1.15^2 + 1.15 * 0.15 + 0.15^2)] = 1
This cancellation is a crucial step in arriving at the final answer. By recognizing and canceling out common factors, we have significantly simplified the expression, leading to a straightforward result. This step highlights the importance of looking for opportunities to simplify expressions by canceling out common terms. The ability to identify and exploit such cancellations is a key skill in algebraic manipulation. The simplification process not only makes the calculation easier but also reveals the underlying structure of the expression. In this case, the cancellation of common terms leads to a remarkably simple result, demonstrating the elegance and efficiency of algebraic techniques.
The Final Answer
After simplifying the expression step by step, we have arrived at the final answer. By applying algebraic identities and canceling out common terms, we have shown that:
(1.15^3 - 0.15^3) / (1.35^2 - 0.35^2) * 1.7 / (1.15^2 + 1.15 * 0.15 + 0.15 * 0.15) = 1
Therefore, the value of the expression is 1. This result is a testament to the power of algebraic simplification techniques. What initially appeared to be a complex expression has been reduced to a simple numerical value through the strategic application of algebraic identities and the careful cancellation of common factors. This process underscores the importance of mastering fundamental algebraic skills for solving mathematical problems. The ability to recognize patterns, apply appropriate identities, and simplify expressions is crucial for success in algebra and beyond. The final answer, 1, is not only a numerical result but also a demonstration of the elegance and efficiency of mathematical methods. This problem serves as a valuable example of how complex expressions can be simplified through systematic application of algebraic principles, highlighting the beauty and power of mathematics.
Conclusion
In conclusion, we have successfully evaluated the value of the given expression by systematically applying algebraic identities and simplifying the terms. The expression, which initially appeared complex, was methodically broken down into smaller, more manageable parts. We utilized the difference of cubes and difference of squares identities to simplify the numerator and the denominator of the fractions involved. This approach allowed us to transform the expression into a form where we could easily identify and cancel out common factors. The strategic application of these algebraic techniques led us to the final answer of 1. This result demonstrates the power of algebraic manipulation in simplifying complex expressions and arriving at concise solutions. The process involved not only the application of specific identities but also a keen eye for recognizing patterns and opportunities for simplification. This is a hallmark of effective problem-solving in mathematics. Furthermore, this exercise highlights the importance of understanding fundamental algebraic concepts and being able to apply them skillfully. The ability to simplify expressions is a crucial skill in mathematics and has wide-ranging applications in various fields. By mastering these techniques, one can approach complex problems with confidence and solve them efficiently. The journey from the initial complex expression to the final simple answer underscores the elegance and power of mathematical reasoning and algebraic manipulation.