Simplifying And Evaluating Algebraic Expressions A Step-by-Step Guide

Algebraic expressions can often appear daunting with their mix of numbers, variables, and operations. But with the right strategies, even the most complex expressions can be simplified. Simplification not only makes the expression easier to understand but also facilitates its evaluation. Let's embark on a journey to dissect and simplify the expression provided, revealing its underlying structure and numerical value.

The cornerstone of simplifying algebraic expressions is recognizing and applying fundamental algebraic identities. These identities, such as the difference of squares, the square of a binomial, and the cube of a binomial, provide shortcuts for factoring and expanding expressions. Mastering these identities is crucial for efficient simplification. In our case, we'll leverage the difference of squares and the expansion of squared terms to unravel the expression's intricacies.

The first step in simplification often involves identifying common factors or patterns within the expression. This might involve factoring out a common term, recognizing a difference of squares, or spotting a perfect square trinomial. Careful observation is key here, as spotting these patterns early can save significant time and effort. Once identified, these patterns can be used to rewrite the expression in a more manageable form.

Once we have factored or expanded the expression, the next step is to combine like terms. This involves adding or subtracting terms with the same variables and exponents. This process often reveals further simplification opportunities, as terms may cancel out or combine to form simpler expressions. It's like tidying up a cluttered room – grouping similar items together makes the overall space more organized and easier to navigate. This process streamlines the expression, reducing its complexity and paving the way for evaluation.

In our specific example, we'll encounter terms involving squares and products of numbers. By carefully expanding and factoring these terms, we can reveal cancellations and simplifications that would not be immediately obvious in the original expression. This methodical approach, breaking down the expression into smaller, more manageable parts, is the key to successful simplification.

Our task is to simplify and evaluate the following expression:

$$\frac{1.9^2 - 1.2 \cdot 1.9 + 0.6^2}{1.9^2 - 0.6^2} \cdot \frac{(2.1 + 1.9)^2 - 2.1 \cdot 1.9}{2.1^3 - 1.9^3}$$

This expression appears complex at first glance, but by breaking it down into smaller parts and applying algebraic identities, we can simplify it considerably. The expression consists of two main fractions, each with its own numerator and denominator. Our strategy will be to simplify each fraction separately before combining them.

Let's begin by examining the first fraction:

$$\frac{1.9^2 - 1.2 \cdot 1.9 + 0.6^2}{1.9^2 - 0.6^2}$$

The numerator resembles a quadratic expression, while the denominator is a difference of squares. This is a crucial observation, as it suggests we can use algebraic identities to factor both parts of the fraction. The difference of squares identity, a² - b² = (a + b)(a - b), will be particularly useful for simplifying the denominator.

Next, we'll turn our attention to the second fraction:

$$\frac{(2.1 + 1.9)^2 - 2.1 \cdot 1.9}{2.1^3 - 1.9^3}$$

The numerator involves a squared term and a product, while the denominator is a difference of cubes. Again, algebraic identities will be our primary tool for simplification. The difference of cubes identity, a³ - b³ = (a - b)(a² + ab + b²), will be essential for factoring the denominator. By recognizing these patterns, we can transform the expression into a more manageable form.

Before diving into the algebraic manipulations, let's take a moment to appreciate the structure of the expression. It's a product of two fractions, each representing a ratio of potentially complex terms. Our goal is to reduce this complexity by identifying and canceling common factors, ultimately arriving at a simplified expression that can be easily evaluated.

Let's embark on the simplification process, tackling each part of the expression systematically.

First Fraction: Numerator

The numerator of the first fraction is: 1.9² - 1.2 * 1.9 + 0.6². While it might not immediately appear so, this expression is a perfect square trinomial. To see this, let's rewrite the middle term: 1.2 * 1.9 = 2 * 0.6 * 1.9. Now the numerator looks like: 1.9² - 2 * 0.6 * 1.9 + 0.6². This is in the form a² - 2ab + b², which factors as (a - b)². Therefore, the numerator simplifies to (1.9 - 0.6)² = 1.3².

First Fraction: Denominator

The denominator of the first fraction is: 1.9² - 0.6². This is a classic difference of squares, which factors as (1.9 + 0.6)(1.9 - 0.6). This simplifies to 2.5 * 1.3.

Putting the First Fraction Together

Now we can rewrite the first fraction as:$\frac{1.3^2}{2.5 \cdot 1.3} = \frac{1.3}{2.5}$

We've already made significant progress by factoring and canceling common factors.

Second Fraction: Numerator

The numerator of the second fraction is: (2.1 + 1.9)² - 2.1 * 1.9. First, simplify the term inside the parentheses: (2.1 + 1.9) = 4. So the numerator becomes: 4² - 2.1 * 1.9 = 16 - 3.99 = 12.01.

Second Fraction: Denominator

The denominator of the second fraction is: 2.1³ - 1.9³. This is a difference of cubes, which factors as (2.1 - 1.9)(2.1² + 2.1 * 1.9 + 1.9²). This simplifies to 0.2 * (4.41 + 3.99 + 3.61) = 0.2 * 12.01.

Putting the Second Fraction Together

Now we can rewrite the second fraction as:$\frac{12.01}{0.2 \cdot 12.01} = \frac{1}{0.2} = 5$

Again, we've simplified significantly by factoring and canceling.

Now that we've simplified both fractions, we can combine them:

$$\frac{1.3}{2.5} \cdot 5$$

To make the calculation easier, let's convert the decimal to a fraction: 2.5 = 5/2. So the expression becomes:

$$\frac{1.3}{\frac{5}{2}} \cdot 5 = 1.3 \cdot \frac{2}{5} \cdot 5$$

The 5s cancel out, leaving us with: 1.3 * 2 = 2.6

Therefore, the simplified value of the expression is 2.6.

This exercise demonstrates the power of algebraic identities in simplifying complex expressions. By recognizing patterns like the difference of squares, the perfect square trinomial, and the difference of cubes, we were able to factor and cancel terms, ultimately arriving at a concise and easily evaluated result. The key takeaway is that a systematic approach, combined with a solid understanding of algebraic identities, can transform seemingly intractable expressions into manageable ones.

In conclusion, simplifying algebraic expressions is not just about finding the numerical answer; it's about understanding the underlying structure and relationships within the expression. This understanding allows us to manipulate the expression strategically, revealing its hidden simplicity and beauty. The ability to simplify expressions is a fundamental skill in mathematics, with applications ranging from solving equations to calculus and beyond. By mastering these techniques, we empower ourselves to tackle a wide range of mathematical challenges with confidence and clarity.