Solving For The Square Root Of 1/64 A Step By Step Guide

When dealing with mathematical problems, especially those involving square roots and fractions, it's important to have a solid understanding of the fundamental principles. In this article, we will explore the process of finding the square root of a fraction and discuss the concept of real numbers in the context of square roots. Let's consider the problem at hand: A. $-\sqrt{\frac{1}{64}} = \square$ and B. The root is not a real number. To solve this, we need to delve into the properties of square roots and fractions. Square roots can sometimes be tricky, especially when dealing with fractions or negative numbers under the radical. A square root of a number x is a value that, when multiplied by itself, gives x. For instance, the square root of 9 is 3 because 3 * 3 = 9. However, when we introduce fractions, the process remains conceptually the same but requires careful handling of the numerator and the denominator. The fraction $\frac{1}{64}$ represents one part of sixty-four equal parts. When we seek its square root, we are essentially looking for a number that, when squared, equals $\frac{1}{64}$. Now, let's break down the square root of $\frac{1}{64}$. The square root of a fraction can be found by taking the square root of the numerator and the square root of the denominator separately. In this case, we have $\sqrt{\frac{1}{64}}$. The square root of the numerator, which is 1, is simply 1 because 1 * 1 = 1. The square root of the denominator, which is 64, is 8 because 8 * 8 = 64. Therefore, $\sqrt{\frac{1}{64}} = \frac{\sqrt{1}}{\sqrt{64}} = \frac{1}{8}$.

Applying the Negative Sign

The problem statement includes a negative sign in front of the square root: $-\sqrt{\frac{1}{64}}$. This means we need to apply the negative sign to our result. We've already determined that $\sqrt{\frac{1}{64}} = \frac{1}{8}$, so $-\sqrt{\frac{1}{64}} = -\frac{1}{8}$. This is a crucial step, as the negative sign changes the value of the expression. It’s essential to remember that the negative sign applies to the entire square root, not just part of it. In the context of the given problem, this step directly addresses the question posed in part A, providing a numerical answer that satisfies the conditions of the expression.

Real Numbers and Square Roots

The second part of the problem, B, asks whether the root is not a real number. To answer this, we need to understand what real numbers are and how they relate to square roots. Real numbers include all numbers that can be represented on a number line. This encompasses rational numbers (such as integers, fractions, and terminating or repeating decimals) and irrational numbers (such as $\sqrt{2}$ and π). Real numbers also include positive and negative numbers, as well as zero. When we talk about square roots, we need to consider the numbers under the radical sign. The square root of a positive number is a real number. For example, $\sqrt{9} = 3$, and 3 is a real number. The square root of zero is also a real number, as $\sqrt{0} = 0$. However, the square root of a negative number is not a real number. This is because there is no real number that, when multiplied by itself, yields a negative result. For instance, $\sqrt{-4}$ is not a real number because no real number squared equals -4. These numbers are classified as imaginary numbers, which are part of the complex number system. In our problem, we are dealing with $-\sqrt{\frac{1}{64}}$. The number under the square root, $\frac{1}{64}$, is positive. Therefore, the square root of $\frac{1}{64}$ is a real number. The negative sign in front of the square root simply changes the sign of the result, but it does not make the number non-real. Thus, $-\frac{1}{8}$ is a real number. Understanding the distinction between real and non-real numbers is crucial in mathematics, especially when dealing with algebraic equations and functions. The concept of real numbers forms the foundation for many mathematical operations, and recognizing when a number falls outside this category is essential for accurate problem-solving.

Conclusion: Solving the Problem

Now, let's revisit the original problem. We have A. $-\sqrt{\frac{1}{64}} = \square$ and B. The root is not a real number. We've determined that $-\sqrt{\frac{1}{64}} = -\frac{1}{8}$. Therefore, the correct answer for A is $-\frac{1}{8}$. As for B, we've established that the root is a real number because $\frac{1}{64}$ is positive, and the negative sign only affects the sign of the result, not its reality. Thus, statement B is incorrect. In summary, solving problems involving square roots and fractions requires a clear understanding of mathematical principles and careful application of these principles. By breaking down the problem into smaller steps, such as finding the square root of the fraction and then applying the negative sign, we can arrive at the correct solution. Additionally, understanding the concept of real numbers and how they relate to square roots is essential for accurately classifying numbers and solving mathematical problems.

When addressing mathematical problems, particularly those involving square roots, it is crucial to ensure a comprehensive understanding of the underlying concepts. In this context, we are tasked with selecting the correct answer from a given set of options concerning the square root of a fraction. The problem presented is: A. $-\sqrt{\frac{1}{64}} = \square$ and B. The root is not a real number. This requires a detailed examination of how square roots of fractions are computed and the distinction between real and non-real numbers. The concept of a square root is fundamental in mathematics. A square root of a number x is a value that, when multiplied by itself, yields x. For instance, the square root of 25 is 5 because 5 multiplied by 5 equals 25. When dealing with fractions, the same principle applies, but it necessitates a careful consideration of both the numerator and the denominator. Specifically, to find the square root of a fraction, one must compute the square root of the numerator and the square root of the denominator separately. This approach simplifies the process and allows for a more methodical solution.

Calculating the Square Root of the Fraction

In our problem, the fraction under consideration is $\frac{1}{64}$. To compute its square root, we need to find the square root of both the numerator (1) and the denominator (64). The square root of 1 is 1, as 1 multiplied by itself is 1. The square root of 64 is 8, because 8 multiplied by 8 equals 64. Therefore, the square root of $\frac{1}{64}$ is $\frac{1}{8}$. This is a critical step in solving the problem, as it provides the basic numerical value before considering the negative sign. This decomposition of the fraction's square root into its numerator and denominator components is a practical technique that simplifies the computation and reduces the likelihood of errors. By addressing each part of the fraction independently, the process becomes more manageable and straightforward. Understanding this method is not only useful for this specific problem but also for a wide array of mathematical scenarios involving square roots and fractions. It exemplifies a systematic approach to problem-solving, which is a valuable skill in mathematics and other disciplines.

Addressing the Negative Sign

The expression in the problem, however, includes a negative sign in front of the square root: $-\sqrt{\frac{1}{64}}$. This means that the result of the square root calculation must be multiplied by -1. We've already established that $\sqrt{\frac{1}{64}} = \frac{1}{8}$, so $-\sqrt{\frac{1}{64}} = -\frac{1}{8}$. The negative sign thus transforms the positive square root into its negative counterpart. This step is a crucial component of the solution, as it accounts for the negative sign present in the original expression. Overlooking the negative sign would lead to an incorrect answer. The inclusion of the negative sign in mathematical expressions often requires careful attention, as it can significantly alter the outcome. In the context of square roots, it is essential to recognize that the negative sign applies to the entire result of the square root operation, not just a portion of it. Therefore, the correct evaluation of the expression necessitates a comprehensive understanding of how negative signs interact with square roots.

Real Numbers Versus Non-Real Numbers

The second part of the problem, B, poses the question of whether the root is not a real number. This requires us to delve into the realm of real numbers and their properties. Real numbers encompass all numbers that can be represented on a number line, including positive numbers, negative numbers, zero, fractions, and irrational numbers such as π and $\sqrt{2}$. In contrast, non-real numbers, also known as imaginary numbers, arise when taking the square root of a negative number. For instance, $\sqrt{-1}$ is denoted as i and is not a real number. The distinction between real and non-real numbers is fundamental in mathematics, particularly in algebra and calculus. Understanding this distinction is crucial for correctly interpreting mathematical expressions and solving equations. In the context of our problem, we are dealing with $-\sqrt{\frac{1}{64}}$. The number under the square root, $\frac{1}{64}$, is positive. Therefore, its square root, $\frac{1}{8}$, is a real number. The negative sign in front of the square root merely changes the sign of the result, making it negative, but it does not render the number non-real. Consequently, $-\frac{1}{8}$ is a real number. To further clarify, any positive number under a square root will yield a real number result. The presence of a negative sign outside the square root does not alter this fact. It is only when the number under the square root is negative that the result becomes a non-real or imaginary number. This understanding is vital for accurate problem-solving and for avoiding common mistakes in mathematical computations.

Conclusion: Identifying the Correct Choice

In summary, to address the original problem, we first computed the square root of $\frac{1}{64}$, which is $\frac{1}{8}$. We then applied the negative sign, resulting in $-\frac{1}{8}$. This corresponds to the correct answer for part A. For part B, we determined that the root is indeed a real number, as the number under the square root was positive. Therefore, the statement "The root is not a real number" is incorrect. This comprehensive approach to the problem, which involves breaking down the expression into manageable steps and considering the properties of real numbers, is essential for accurate mathematical problem-solving. It exemplifies the importance of not only knowing the procedures but also understanding the underlying concepts and principles. By carefully examining each component of the problem and applying the relevant mathematical rules, we can confidently arrive at the correct solution. This methodical approach is a valuable skill that extends beyond mathematics and into various aspects of analytical thinking and problem-solving.

Mathematical problems often require a systematic approach to unravel their complexities. When we encounter expressions involving square roots, it's essential to have a clear understanding of the rules and properties governing these operations. In this article, we aim to break down the process of solving a specific problem: A. $-\sqrt{\frac{1}{64}} = \square$ and B. The root is not a real number. This problem combines the concepts of square roots, fractions, and real numbers, making it a comprehensive exercise in mathematical reasoning. To tackle this, we'll dissect each component and provide a step-by-step solution. The first concept to grasp is the meaning of a square root. A square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 16 is 4 because 4 * 4 = 16. When dealing with fractions under a square root, we apply the same principle, but we must consider both the numerator and the denominator separately. This approach simplifies the calculation and helps in understanding the underlying mathematics.

Calculating the Square Root of a Fraction

In our problem, we have the fraction $\frac{1}{64}$ under the square root. To find the square root of this fraction, we need to determine the square root of the numerator (1) and the square root of the denominator (64). The square root of 1 is straightforward, as 1 multiplied by itself equals 1. Therefore, $\sqrt{1} = 1$. The square root of 64 is also a common value to recognize. Since 8 multiplied by 8 equals 64, we have $\sqrt{64} = 8$. Now, we can express the square root of the fraction as the square root of the numerator divided by the square root of the denominator: $\sqrt{\frac{1}{64}} = \frac{\sqrt{1}}{\sqrt{64}} = \frac{1}{8}$. This step is crucial because it simplifies the original expression and provides a clear numerical value. By breaking down the problem into smaller, more manageable parts, we can avoid confusion and ensure accuracy. The process of finding the square root of a fraction is a fundamental skill in mathematics, applicable in various contexts, from algebra to calculus. Understanding this process not only helps in solving this particular problem but also builds a solid foundation for more complex mathematical operations.

Applying the Negative Sign to the Result

The problem statement, however, includes a negative sign in front of the square root: $-\sqrt{\frac{1}{64}}$. This means that we need to apply this negative sign to the result we obtained in the previous step. We found that $\sqrt{\frac{1}{64}} = \frac{1}{8}$, so now we apply the negative sign: $-\sqrt{\frac{1}{64}} = -\frac{1}{8}$. This step is essential because it directly addresses the specific expression given in the problem. The negative sign changes the value from a positive fraction to a negative fraction. It's a simple but crucial detail that must not be overlooked. When working with mathematical expressions, it's always important to pay close attention to signs, as they can significantly impact the final answer. A common mistake is to disregard the negative sign or apply it incorrectly. By carefully considering the negative sign, we ensure that our solution accurately reflects the original problem. This attention to detail is a hallmark of effective mathematical problem-solving.

Understanding Real Numbers and Square Roots

The second part of the problem, B, asks whether the root is not a real number. To answer this question, we need to understand the concept of real numbers and how they relate to square roots. Real numbers include all numbers that can be represented on a number line. This encompasses positive numbers, negative numbers, zero, fractions, decimals, and irrational numbers (such as $\sqrt{2}$ and π). Real numbers are the foundation of much of mathematics, and they are used in countless applications. In contrast, non-real numbers, also known as imaginary numbers, arise when we take the square root of a negative number. For example, $\sqrt{-1}$ is not a real number; it is denoted by the symbol i and is part of the complex number system. When we have a positive number under a square root, the result is always a real number. In our problem, the number under the square root, $\frac{1}{64}$, is positive. Therefore, the square root of $\frac{1}{64}$ is a real number. The negative sign in front of the square root does not change this fact; it simply makes the result negative, but it remains a real number. Thus, $-\frac{1}{8}$ is a real number. This understanding is crucial for correctly classifying numbers and for solving equations and inequalities. Knowing the difference between real and non-real numbers is a fundamental aspect of mathematical literacy. It enables us to make accurate judgments about the nature of solutions and to avoid common pitfalls in mathematical reasoning.

Conclusion: Choosing the Correct Answer

To summarize, we have systematically solved the problem A. $-\sqrt{\frac{1}{64}} = \square$ and B. The root is not a real number. We found that $-\sqrt{\frac{1}{64}} = -\frac{1}{8}$, which is the answer for part A. For part B, we determined that the root is a real number, so the statement "The root is not a real number" is incorrect. Therefore, the correct choice is A. $-\frac{1}{8}$, and B is false. This step-by-step solution demonstrates the importance of breaking down complex problems into smaller, more manageable parts. By addressing each component individually and applying the relevant mathematical rules, we can arrive at the correct answer with confidence. This approach is not only effective for this particular problem but also serves as a model for solving a wide range of mathematical challenges. The ability to dissect a problem, identify the key concepts, and apply appropriate techniques is a valuable skill that extends beyond the realm of mathematics and into various aspects of analytical thinking and problem-solving.