Simplifying (-9 - √24) / (2(2)) A Step-by-Step Guide

This comprehensive guide breaks down the simplification of the mathematical expression (-9 - √24) / (2(2)). We will explore each step, from understanding the initial expression to arriving at the most simplified form. Whether you're a student brushing up on your algebra skills or someone looking to enhance your mathematical understanding, this guide provides a clear and detailed explanation.

Breaking Down the Initial Expression

The given expression is (-9 - √24) / (2(2)). To simplify this, we will address the square root and the denominator separately before combining the terms. The expression involves a fraction, a square root, and basic arithmetic operations. The first step is to simplify the square root term, √24. The ability to simplify square roots is crucial in algebra, as it allows us to express numbers in their simplest radical form. This not only makes the numbers easier to work with but also provides a clearer understanding of their magnitude. Simplification often involves identifying perfect square factors within the radicand (the number inside the square root). For example, in the case of √24, we look for perfect squares that divide 24.

Simplifying the Square Root

The square root of 24 (√24) can be simplified by finding the prime factorization of 24. The prime factorization of 24 is 2 x 2 x 2 x 3, which can be written as 2^3 x 3. To simplify the square root, we look for pairs of factors, as each pair can be taken out of the square root as a single factor. In this case, we have a pair of 2s. Therefore, √24 can be written as √(2^2 x 2 x 3). The square root of 2^2 is 2, so we can rewrite the expression as 2√(2 x 3), which simplifies to 2√6. This process of simplifying square roots is essential in algebra for making expressions easier to manipulate and understand. It also helps in comparing and combining like terms in more complex expressions. By reducing the square root to its simplest form, we can more easily perform other operations, such as addition, subtraction, multiplication, and division.

Addressing the Denominator

The denominator of the expression is 2(2), which simplifies to 4. This is a straightforward arithmetic operation, but it's important to address it early in the simplification process. A clear understanding of the denominator is crucial because it affects how we handle the entire expression. Simplifying the denominator often makes the subsequent steps, such as dividing or factoring, much easier. In this case, multiplying 2 by 2 gives us 4, which is a simple whole number. This simplified denominator will be used in the next steps to further simplify the entire expression. It’s a small step, but it's a necessary one to ensure we're working with the most basic form of the expression.

Combining the Simplified Terms

Now that we've simplified √24 to 2√6 and the denominator 2(2) to 4, we can rewrite the original expression as (-9 - 2√6) / 4. This form is much simpler than the original, but we can further refine it. We have a fraction with a binomial in the numerator and a single term in the denominator. To simplify this, we can split the fraction into two separate fractions, each with the same denominator. This allows us to handle each term in the numerator individually and potentially simplify them further. This step is a common technique in algebra for dealing with fractions that have binomials or polynomials in the numerator.

Splitting the Fraction

Splitting the fraction (-9 - 2√6) / 4 into two separate fractions gives us -9/4 - (2√6)/4. This step is based on the principle that (a + b) / c = a/c + b/c. By splitting the fraction, we can simplify each term individually. The first term, -9/4, is already in its simplest form as an improper fraction. The second term, (2√6)/4, can be simplified further by reducing the fraction. This involves looking for common factors between the numerator and the denominator. In this case, both the numerator and the denominator have a factor of 2, which can be cancelled out. Splitting fractions is a useful technique not only for simplifying expressions but also for solving equations and integrating functions in calculus. It allows us to break down complex problems into smaller, more manageable parts.

Simplifying the Second Term

The second term, (2√6)/4, can be simplified by dividing both the numerator and the denominator by their common factor, which is 2. Dividing 2√6 by 2 gives us √6, and dividing 4 by 2 gives us 2. So, the simplified second term is √6/2. This simplification is an example of reducing a fraction to its lowest terms, a fundamental skill in algebra and arithmetic. By simplifying fractions, we make them easier to work with and understand. This process involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number. In this case, the GCD of 2 and 4 is 2, so we divide both by 2 to get the simplified fraction. Simplifying fractions is crucial for solving equations, comparing values, and performing other mathematical operations accurately.

Final Simplified Expression

After simplifying both terms, the expression becomes -9/4 - √6/2. This is the most simplified form of the original expression, (-9 - √24) / (2(2)). We have addressed the square root, simplified the denominator, split the fraction, and reduced each term to its simplest form. This final expression is easier to work with and understand than the original. It consists of two terms: a simple fraction (-9/4) and a fraction involving a square root (-√6/2). This form allows for easier comparison with other expressions and can be readily used in further calculations or algebraic manipulations. The process of simplifying expressions is a core skill in mathematics, essential for success in algebra, calculus, and beyond. It involves a combination of arithmetic operations, understanding of square roots, and fraction manipulation.

Expressing with a Common Denominator (Optional)

While -9/4 - √6/2 is the simplified form, we can optionally express it with a common denominator to combine the terms into a single fraction. To do this, we need to find the least common multiple (LCM) of the denominators, which are 4 and 2. The LCM of 4 and 2 is 4. We then convert each fraction to have the denominator 4. The first term, -9/4, already has the denominator 4. For the second term, -√6/2, we multiply both the numerator and the denominator by 2 to get -2√6/4. Now, we can combine the two fractions:

(-9/4) - (2√6/4) = (-9 - 2√6) / 4

This gives us the single fraction (-9 - 2√6) / 4, which is an alternative way to express the simplified form. While it may not always be necessary to express the final answer as a single fraction, it can be useful in certain contexts, such as when adding or subtracting multiple fractions. It also provides a different perspective on the expression and can sometimes reveal further simplifications or relationships.

Conclusion

Simplifying the expression (-9 - √24) / (2(2)) involves several steps, including simplifying the square root, addressing the denominator, splitting the fraction, and reducing each term to its simplest form. The final simplified expression is -9/4 - √6/2, or optionally, (-9 - 2√6) / 4. This process demonstrates the importance of breaking down complex problems into smaller, more manageable steps. Each step, from simplifying the square root to combining fractions, requires a solid understanding of basic mathematical principles. By mastering these techniques, you can confidently tackle more complex algebraic expressions and problems. Simplification is not just about finding the correct answer; it's about developing a deeper understanding of mathematical structures and relationships.

This comprehensive guide has provided a detailed walkthrough of the simplification process, highlighting the key concepts and techniques involved. Whether you're a student, a teacher, or simply someone with an interest in mathematics, this guide serves as a valuable resource for understanding and simplifying algebraic expressions.