Simplifying (-3)^2 + √(48) Rounded To Two Decimal Places

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Understanding the Basics

In this comprehensive guide, we will delve into the simplification of the mathematical expression (-3)^2 + √48, meticulously breaking down each step to ensure a clear understanding. This expression combines basic arithmetic operations with the concept of square roots, making it a fantastic example for grasping fundamental mathematical principles. Our goal is to not only find the solution but also to provide an in-depth explanation that enhances your problem-solving skills. The expression involves two primary components: the square of a negative number and the square root of a number. Before we dive into the specifics, let's briefly review these concepts. Squaring a number means multiplying it by itself. For instance, the square of -3, denoted as (-3)^2, is the result of multiplying -3 by -3. Square roots, on the other hand, are the inverse operation of squaring a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. Understanding these concepts is crucial for simplifying the given expression. Our approach will be methodical, ensuring that each step is thoroughly explained. We'll begin by addressing the squared term, then simplify the square root, and finally, combine the results to arrive at the final answer. We will also discuss how to round off the answer to the nearest two decimal places, a practical skill in mathematics. This process will not only provide you with the solution to this specific problem but also equip you with the knowledge to tackle similar mathematical challenges with confidence. So, let's embark on this mathematical journey and simplify the expression (-3)^2 + √48 together.

Step-by-Step Simplification

To begin, let's address the first part of the expression: (-3)^2. This signifies -3 multiplied by itself. When we multiply two negative numbers, the result is a positive number. Therefore, (-3) * (-3) = 9. This is a fundamental rule in mathematics that we must remember to avoid errors. Next, we turn our attention to the second part of the expression: √48. Simplifying square roots often involves finding perfect square factors within the number under the root. A perfect square is a number that can be obtained by squaring an integer. For example, 4, 9, 16, and 25 are perfect squares because they are the squares of 2, 3, 4, and 5, respectively. To simplify √48, we look for the largest perfect square that divides 48. In this case, the largest perfect square factor of 48 is 16, since 48 = 16 * 3. Now, we can rewrite √48 as √(16 * 3). Using the property of square roots that states √(a * b) = √a * √b, we can further simplify this as √16 * √3. We know that √16 = 4, so our expression becomes 4√3. This simplified form is much easier to work with. Now that we have simplified both parts of the expression, we can combine them. We have (-3)^2 = 9 and √48 = 4√3. Adding these together, we get 9 + 4√3. This is the exact simplified form of the expression. However, to get a numerical approximation, we need to estimate the value of √3. The square root of 3 is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation goes on infinitely without repeating. A common approximation for √3 is 1.732. We will use this approximation to find a decimal value for our expression. So, 4√3 is approximately 4 * 1.732 = 6.928. Adding this to 9, we get 9 + 6.928 = 15.928. Finally, we need to round this result to the nearest two decimal places, as specified in the problem.

Rounding to Two Decimal Places

Rounding to two decimal places means we want to keep only two digits after the decimal point. To do this, we look at the third digit after the decimal point. If this digit is 5 or greater, we round up the second digit. If it is less than 5, we leave the second digit as it is. In our case, we have the number 15.928. The third digit after the decimal point is 8, which is greater than 5. Therefore, we need to round up the second digit. The second digit after the decimal point is 2. Rounding it up by one, we get 3. So, 15.928 rounded to two decimal places is 15.93. This is our final answer. Rounding is a crucial skill in mathematics, especially when dealing with irrational numbers like square roots that have infinite decimal expansions. It allows us to provide practical, usable approximations of values. The rule for rounding is simple but essential: look at the next digit after the place you are rounding to. If it's 5 or more, round up; if it's less than 5, round down. In the context of the given problem, rounding to two decimal places provides a balance between precision and simplicity. While the exact value of 9 + 4√3 is an irrational number, 15.93 gives a close approximation that is easy to understand and use in practical applications. This step-by-step explanation not only provides the answer but also highlights the importance of understanding the underlying mathematical principles and techniques. From simplifying square roots to rounding decimals, each step is crucial for arriving at the correct solution. By mastering these skills, you will be well-equipped to tackle a wide range of mathematical problems with confidence. In summary, the process of rounding is an essential part of mathematical problem-solving, allowing us to express complex numbers in a more manageable and practical form. Always remember to consider the context of the problem when deciding how many decimal places to round to, as the appropriate level of precision may vary depending on the situation.

Final Answer

Therefore, simplifying the expression (-3)^2 + √48 and rounding the result to the nearest two decimal places gives us 15.93. This result is obtained by first calculating (-3)^2 which equals 9. Then, we simplified √48 to 4√3, which is approximately 4 * 1.732 = 6.928. Adding these two values together, we get 9 + 6.928 = 15.928. Finally, rounding 15.928 to two decimal places gives us the final answer of 15.93. This comprehensive solution demonstrates the step-by-step process of simplifying mathematical expressions involving squares, square roots, and rounding. Each step is crucial in ensuring accuracy and understanding the underlying mathematical principles. The ability to simplify such expressions is fundamental in algebra and calculus, making this a valuable skill for students and professionals alike. Furthermore, the process of rounding to a specific number of decimal places is an essential practical skill. It allows us to express numbers in a more manageable and understandable format, especially when dealing with irrational numbers that have infinite decimal expansions. In this case, rounding to two decimal places provides a precise yet practical approximation of the final answer. The journey to this final answer involved several key mathematical concepts and techniques. Understanding the order of operations, simplifying square roots, and applying rounding rules are all critical components of this process. By mastering these skills, one can confidently approach a wide range of mathematical problems. The solution to this problem not only provides a numerical answer but also reinforces the importance of methodical problem-solving. Breaking down a complex expression into smaller, more manageable parts is a strategy that can be applied to many areas of mathematics and beyond. Each step builds upon the previous one, leading to a clear and accurate solution. In conclusion, the simplification of (-3)^2 + √48, rounded to two decimal places, is 15.93. This result is achieved through a careful and methodical application of mathematical principles and techniques. The process highlights the importance of understanding squares, square roots, simplification, and rounding, all of which are essential skills in mathematics.