Solving Mathematical Problem 1.1 M = √(a - 3bc) A Step By Step Explanation

11.1.1 Calculating the Value of m for a = -6, b = -2, and c = 5

In this section, we embark on a journey to decipher the value of m when given specific values for the variables a, b, and c within the expression m = √(a - 3bc). The given values are a = -6, b = -2, and c = 5. Our mission is to substitute these values into the expression, perform the necessary calculations, and express the final answer in its simplest root form. To achieve this, we will meticulously follow the order of operations, ensuring accuracy and precision at every step. The first step involves substituting the given values into the expression: m = √(-6 - 3(-2)(5)). Next, we simplify the expression within the square root by performing the multiplication: m = √(-6 - (-30)). Remember that multiplying two negative numbers results in a positive number. The expression then becomes m = √(-6 + 30). Now, we perform the addition within the square root: m = √(24). Our final step is to simplify the square root. We look for the largest perfect square that divides 24, which is 4. We can rewrite √(24) as √(4 * 6). Using the property of square roots that √(ab) = √a * √b, we get √(4 * 6) = √4 * √6. Since √4 = 2, the simplified expression becomes m = 2√6. Therefore, the value of m, when a = -6, b = -2, and c = 5, expressed in its simplest root form, is 2√6. This meticulous approach ensures we arrive at the correct solution while adhering to mathematical principles.

This problem exemplifies the importance of following the order of operations (PEMDAS/BODMAS) when evaluating mathematical expressions. Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right) dictate the sequence in which operations must be performed. By adhering to this order, we avoid errors and arrive at accurate results. In this specific problem, we first performed the multiplication within the square root before proceeding with the addition. This ensured that we correctly calculated the value of the expression under the square root before taking the square root itself. Furthermore, the simplification of the square root involved identifying the largest perfect square factor of the radicand (the number under the square root). This allowed us to express the square root in its simplest form, which is a crucial step in providing a complete and accurate solution. The ability to simplify square roots is a fundamental skill in algebra and is essential for solving various mathematical problems. In summary, this problem highlights the significance of meticulous calculation, adherence to the order of operations, and the ability to simplify expressions to arrive at the correct solution. The final answer, 2√6, represents the value of m in its most simplified form.

11.1.2 Pinpointing Consecutive Integers Enclosing the Value of m

Here, our objective shifts from finding the exact value of m to locating the two consecutive integers between which m resides. We already know that m = 2√6. To determine the consecutive integers, we need to estimate the value of √6. We know that √4 = 2 and √9 = 3. Since 6 lies between 4 and 9, √6 must lie between 2 and 3. A closer approximation reveals that √6 is approximately 2.45. Now, we multiply this approximation by 2 (from the expression m = 2√6), yielding m ≈ 2 * 2.45 ≈ 4.9. Based on this approximation, we can confidently conclude that the value of m lies between the consecutive integers 4 and 5. To further solidify our understanding, let's consider the squares of these integers. 4 squared is 16, and 5 squared is 25. The square of m, which is (2√6)², is equal to 4 * 6 = 24. Since 24 lies between 16 and 25, it confirms that m indeed lies between 4 and 5.

This approach demonstrates a valuable technique for estimating the value of expressions involving square roots without relying on a calculator. By understanding the perfect squares that surround the radicand (the number under the square root), we can effectively narrow down the range within which the square root lies. This method is particularly useful in situations where an exact value is not required, and an approximation suffices. In this case, we leveraged our knowledge of √4 and √9 to estimate the value of √6. This allowed us to approximate the value of m and subsequently identify the consecutive integers that enclose it. The strategy of squaring the integers and comparing them to the square of m provides an additional layer of verification, ensuring the accuracy of our estimation. Moreover, this problem underscores the importance of number sense and the ability to work with approximations. In many real-world scenarios, precise calculations are not always necessary, and estimations can provide valuable insights. The ability to quickly and accurately estimate values is a crucial skill in mathematics and beyond. In conclusion, by employing approximation techniques and leveraging our understanding of perfect squares, we successfully determined that the value of m lies between the consecutive integers 4 and 5, without explicitly calculating the exact value of m.

This comprehensive analysis of question 11.1 demonstrates a methodical approach to solving mathematical problems. By breaking down the problem into smaller, manageable steps, we can tackle complex expressions with confidence. The ability to substitute values, simplify expressions, and estimate values are fundamental skills that are essential for success in mathematics and related fields. The problem also highlights the importance of understanding the properties of square roots and the order of operations. These concepts are building blocks for more advanced mathematical topics. Through consistent practice and a deep understanding of these principles, students can develop their problem-solving abilities and excel in mathematics. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively to solve problems. By embracing this approach, we can unlock the beauty and power of mathematics and use it to tackle a wide range of challenges.

• Keywords: Square root, expression, simplification, consecutive integers, estimation, order of operations, substitution, mathematical problem-solving

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