Simplifying The Expression (7 - √3)(-6 + √3) A Step-by-Step Guide

In mathematics, simplifying expressions is a fundamental skill. It involves manipulating an expression into a more concise and manageable form. This often makes it easier to understand the expression's value, solve equations, or perform further calculations. This article will delve into the process of simplifying the expression (7 - √3)(-6 + √3). We'll break down the steps involved, explain the underlying mathematical principles, and provide a clear, step-by-step guide to arrive at the simplified answer. Mastering this type of simplification is crucial for success in algebra and beyond, as it forms the basis for more complex mathematical operations.

The ability to simplify expressions is a cornerstone of mathematical proficiency. It's not just about getting the right answer; it's about understanding the underlying structure and relationships within the expression. When we simplify, we are essentially rewriting the expression in a way that makes its inherent properties more apparent. This can involve combining like terms, applying the distributive property, factoring, or rationalizing denominators. In the case of (7 - √3)(-6 + √3), we'll be focusing on the distributive property and the rules for multiplying radicals. Each step in the simplification process is a logical progression, guided by established mathematical principles. The more comfortable you become with these principles, the more confident you'll be in your ability to tackle complex mathematical problems.

The expression (7 - √3)(-6 + √3) presents a specific type of challenge involving both integers and radicals. Radicals, like √3, represent numbers that, when multiplied by themselves, equal the number under the radical sign. Simplifying expressions with radicals often requires careful application of the distributive property and the rules for multiplying radicals. We need to remember that √a * √b = √(a*b) and that the square of a square root cancels out the radical (√a * √a = a). The combination of integers and radicals in this expression necessitates a systematic approach, ensuring that each term is multiplied correctly and that like terms are combined appropriately. The simplification process will not only provide the answer but also reinforce the understanding of how different mathematical concepts interact within a single expression.

Step 1: Applying the Distributive Property (FOIL Method)

To simplify the expression (7 - √3)(-6 + √3), the first step is to apply the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This involves multiplying each term in the first binomial by each term in the second binomial. Let's break this down:

• First: Multiply the first terms of each binomial: 7 * -6 = -42
• Outer: Multiply the outer terms of the binomials: 7 * √3 = 7√3
• Inner: Multiply the inner terms of the binomials: -√3 * -6 = 6√3
• Last: Multiply the last terms of each binomial: -√3 * √3 = -3

So, after applying the distributive property, we have: -42 + 7√3 + 6√3 - 3. This step is crucial because it expands the expression, allowing us to see all the individual terms that need to be combined and simplified. The FOIL method is a systematic way to ensure that every term in the first binomial is multiplied by every term in the second binomial. This eliminates the risk of missing any multiplications and ensures that we have a complete expansion of the original expression. Mastering the distributive property is essential for simplifying algebraic expressions and solving equations.

The distributive property is a fundamental concept in algebra, and its correct application is essential for accurate simplification. In the context of (7 - √3)(-6 + √3), the FOIL method provides a structured approach to ensure that no terms are missed during multiplication. Each multiplication (First, Outer, Inner, Last) contributes to the expanded form of the expression, which then allows us to identify and combine like terms. For instance, multiplying the 'Outer' terms (7 * √3) and the 'Inner' terms (-√3 * -6) results in terms with the radical √3, which can be combined in the subsequent steps. The distributive property is not just a mechanical process; it's a reflection of the fundamental principles of arithmetic and algebra, allowing us to break down complex expressions into simpler components.

Step 2: Combining Like Terms

After applying the distributive property in the expression -42 + 7√3 + 6√3 - 3, the next step is to combine like terms. Like terms are terms that have the same variable and exponent or, in this case, the same radical. We can identify two types of like terms in our expanded expression: the constant terms (-42 and -3) and the terms with the radical √3 (7√3 and 6√3). Combining like terms simplifies the expression by grouping together terms that can be added or subtracted. This step reduces the number of terms in the expression and makes it easier to understand and manipulate.

First, let's combine the constant terms: -42 - 3 = -45. This involves simply adding the two negative numbers together. Next, we combine the terms with the radical √3. To do this, we add the coefficients (the numbers in front of the radical): 7√3 + 6√3 = (7 + 6)√3 = 13√3. The √3 acts like a variable in this case, and we are essentially adding 7 of something (√3) to 6 of the same thing (√3), resulting in 13 of that thing (√3). This process of combining coefficients is analogous to combining like terms in algebraic expressions, where we combine terms with the same variable raised to the same power. The ability to accurately combine like terms is a crucial skill in algebra and beyond, as it allows us to simplify expressions and solve equations more efficiently.

The concept of combining like terms is fundamental to algebraic simplification. It's based on the idea that we can only add or subtract terms that are