Evaluating 0.33 × (√50 - √18) + (√2)/2 A Step-by-Step Guide
Understanding the Expression
At the heart of this mathematical problem lies the need to simplify and evaluate an expression involving square roots and decimal multiplication. Our goal is to break down the expression step by step, ensuring each operation is performed accurately to arrive at the final answer. The expression we are tasked with evaluating is: 0.33 × (√50 - √18) + (√2)/2. This expression combines several mathematical concepts including decimal multiplication, square roots, and basic arithmetic operations. Before we dive into the step-by-step solution, let’s first understand the key components and how they interact with each other.
The expression consists of three main parts: the first part involves multiplying 0.33 by the difference between the square roots of 50 and 18. The second part involves dividing the square root of 2 by 2. And the third part is the addition of the results from the first two parts. To simplify this expression, we will need to address each part individually before combining them. The square roots present in the expression require simplification. We need to express √50 and √18 in their simplest forms by factoring out perfect squares. This will make the subtraction within the parentheses easier to manage. 0.33 is a decimal number, which we might find easier to work with if converted into a fraction. Doing so can help in precise calculations, especially when dealing with square roots. The term (√2)/2 is already in a relatively simplified form, but we need to keep it in mind for the final addition step. The order of operations (PEMDAS/BODMAS) is crucial here. We will first handle the operations within the parentheses, then perform any multiplication or division, and finally, complete the addition. This approach ensures we follow the correct sequence to arrive at the accurate result. Each step in evaluating this expression builds upon the previous one. A clear and methodical approach is essential to avoid errors and ensure accuracy. By understanding these foundational concepts and employing a careful, step-by-step method, we can successfully evaluate the expression and arrive at the correct solution. Let’s proceed with the simplification and evaluation process.
Step-by-Step Solution
To accurately evaluate the expression 0.33 × (√50 - √18) + (√2)/2, we'll proceed methodically, breaking it down into manageable steps. This approach ensures clarity and precision in our calculations. Let's begin by addressing the square roots within the parentheses. We need to simplify √50 and √18 by identifying perfect square factors. Starting with √50, we can express 50 as 25 × 2, where 25 is a perfect square. Therefore, √50 = √(25 × 2) = √25 × √2 = 5√2. Similarly, for √18, we can express 18 as 9 × 2, where 9 is a perfect square. Thus, √18 = √(9 × 2) = √9 × √2 = 3√2. Now that we have simplified the square roots, we can rewrite the expression inside the parentheses: √50 - √18 = 5√2 - 3√2. Subtracting these terms is straightforward since they both contain √2. Combining like terms, we get 5√2 - 3√2 = (5 - 3)√2 = 2√2. Next, we will address the decimal multiplication. The expression now looks like this: 0.33 × (2√2) + (√2)/2. It might be beneficial to convert 0.33 into a fraction for more precise calculations. 0.33 is approximately equal to 1/3. Substituting 1/3 for 0.33, we have (1/3) × (2√2) + (√2)/2. Now, let's perform the multiplication: (1/3) × (2√2) = (2√2)/3. Our expression now is: (2√2)/3 + (√2)/2. To add these two terms, we need a common denominator. The least common multiple of 3 and 2 is 6. So, we will convert both fractions to have a denominator of 6. For the first term, (2√2)/3, we multiply the numerator and denominator by 2: (2√2)/3 × 2/2 = (4√2)/6. For the second term, (√2)/2, we multiply the numerator and denominator by 3: (√2)/2 × 3/3 = (3√2)/6. Now we can add the two terms: (4√2)/6 + (3√2)/6 = (4√2 + 3√2)/6 = (7√2)/6. This is the simplified form of the expression. To get a numerical approximation, we can substitute the value of √2 ≈ 1.414. (7√2)/6 ≈ (7 × 1.414)/6 ≈ 9.898/6 ≈ 1.6497. Therefore, the value of the expression 0.33 × (√50 - √18) + (√2)/2 is approximately 1.6497. By following this step-by-step approach, we have simplified the expression and arrived at the final answer with clarity and accuracy.
Detailed Breakdown of Each Step
To fully grasp the evaluation of the expression 0.33 × (√50 - √18) + (√2)/2, it's essential to dissect each step and understand the mathematical principles behind it. This detailed breakdown will provide a comprehensive understanding of the process. The first critical step is simplifying the square roots: √50 and √18. This involves expressing the numbers under the square root as products of their factors, particularly identifying any perfect square factors. For √50, we recognize that 50 can be factored into 25 × 2. Since 25 is a perfect square (5² = 25), we can rewrite √50 as √(25 × 2). Using the property of square roots that √(a × b) = √a × √b, we get √50 = √25 × √2 = 5√2. Similarly, for √18, we factor 18 into 9 × 2. Since 9 is a perfect square (3² = 9), we can rewrite √18 as √(9 × 2). Applying the same property, √18 = √9 × √2 = 3√2. By simplifying the square roots, we've transformed the expression inside the parentheses into 5√2 - 3√2. This makes the subtraction step much easier. Now, we move on to subtracting the simplified square root terms: 5√2 - 3√2. Since both terms have the same radical part (√2), we can treat them as like terms and subtract their coefficients. This is similar to subtracting algebraic expressions like 5x - 3x. The subtraction yields (5 - 3)√2 = 2√2. The next step involves addressing the decimal multiplication. The expression now looks like 0.33 × (2√2) + (√2)/2. For accuracy, it's often beneficial to convert decimals into fractions, especially when dealing with radicals. 0.33 is approximately equal to 1/3. Substituting 1/3 for 0.33, we get (1/3) × (2√2) + (√2)/2. Performing the multiplication, (1/3) × (2√2) equals (2√2)/3. This step simplifies the expression further, preparing it for the final addition. Now, we focus on adding the two terms: (2√2)/3 and (√2)/2. To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 2 is 6. We convert both fractions to have a denominator of 6. For (2√2)/3, we multiply both the numerator and the denominator by 2: (2√2)/3 × 2/2 = (4√2)/6. For (√2)/2, we multiply both the numerator and the denominator by 3: (√2)/2 × 3/3 = (3√2)/6. With a common denominator, we can now add the two fractions: (4√2)/6 + (3√2)/6. Adding the numerators, we get (4√2 + 3√2)/6. Combining like terms in the numerator, we have (7√2)/6. This is the simplified exact form of the expression. Finally, to obtain a numerical approximation, we substitute the value of √2 ≈ 1.414 into the expression (7√2)/6. This gives us (7 × 1.414)/6 ≈ 9.898/6 ≈ 1.6497. Therefore, the value of the expression 0.33 × (√50 - √18) + (√2)/2 is approximately 1.6497. Each step in this process builds upon the previous one, ensuring a methodical and accurate evaluation of the expression.
Alternative Approaches to Solving the Expression
While the step-by-step method provides a clear and direct approach to evaluating the expression 0.33 × (√50 - √18) + (√2)/2, exploring alternative approaches can offer additional insights and potentially simplify the process. Let's consider some alternative methods that could be employed. One alternative approach involves using decimal approximations earlier in the process. Instead of converting 0.33 to a fraction immediately, we could work with its decimal form throughout. Similarly, we could approximate √50 and √18 as decimals before performing any operations. For instance, √50 ≈ 7.071 and √18 ≈ 4.243. Using these approximations, the expression inside the parentheses becomes 7.071 - 4.243 ≈ 2.828. Multiplying this by 0.33, we get 0.33 × 2.828 ≈ 0.933. The last term, (√2)/2, can be approximated as 1.414/2 ≈ 0.707. Adding these two results, we have 0.933 + 0.707 ≈ 1.640. This approach yields an approximate answer that is close to the exact solution, although it might be less accurate due to rounding errors. Another approach involves factoring out common terms to simplify the expression. We start by simplifying the square roots as before: √50 = 5√2 and √18 = 3√2. The expression inside the parentheses is 5√2 - 3√2 = 2√2. The original expression becomes 0.33 × (2√2) + (√2)/2. We can rewrite 0.33 as 1/3, so the expression is (1/3) × (2√2) + (√2)/2. This can be written as (2√2)/3 + (√2)/2. Now, we can factor out √2 from both terms: √2 × (2/3 + 1/2). Adding the fractions inside the parentheses, we need a common denominator, which is 6. So, 2/3 becomes 4/6, and 1/2 becomes 3/6. The sum is 4/6 + 3/6 = 7/6. Therefore, the expression simplifies to √2 × (7/6), which is (7√2)/6. This matches the exact simplified form we obtained earlier. We can then approximate √2 as 1.414 to get a numerical answer, as before. Additionally, using technology such as calculators or computer algebra systems (CAS) can provide a quick and accurate way to evaluate the expression. These tools can handle square roots and decimal operations with high precision. By inputting the expression directly into a calculator or CAS software, we can obtain the result without manual calculations. This method is particularly useful for complex expressions where manual simplification might be tedious and error-prone. Each of these alternative approaches offers a different way to tackle the problem, providing flexibility and potentially simplifying the evaluation process depending on the specific context and tools available. While the step-by-step method ensures a clear understanding of each operation, these alternatives can be valuable for verifying results or exploring different problem-solving strategies.
Practical Applications and Relevance
Understanding how to evaluate expressions like 0.33 × (√50 - √18) + (√2)/2 extends beyond the classroom and has practical applications in various fields. The ability to simplify and compute such expressions is a fundamental skill in mathematics and science. In physics, for example, calculations involving square roots often arise in the context of mechanics, optics, and electromagnetism. When dealing with kinematic equations, energy calculations, or wave phenomena, square roots are frequently encountered. The ability to simplify these expressions can lead to more manageable calculations and a better understanding of the physical phenomena being modeled. In engineering, evaluating expressions with square roots is crucial in many disciplines. Civil engineers may encounter such expressions when calculating structural loads or analyzing stresses and strains in materials. Electrical engineers might use square roots in circuit analysis, particularly when dealing with impedance or power calculations. Mechanical engineers often work with square roots in thermodynamics, fluid mechanics, and control systems. For example, calculating the natural frequency of a mechanical system often involves square roots. In computer graphics and game development, square roots are used extensively in vector and matrix operations. Calculating distances, normalizing vectors, and performing geometric transformations often require evaluating expressions involving square roots. Efficiently simplifying these expressions can improve performance in graphical applications. In finance, square roots appear in various models and calculations. For example, the standard deviation, a measure of volatility, involves calculating the square root of the variance. Financial analysts use such calculations to assess risk and make investment decisions. In statistics and data analysis, square roots are integral to many statistical measures, such as standard deviation and root mean square error. Being able to work with these expressions is essential for interpreting data and making informed decisions. Beyond these specific fields, the ability to simplify and evaluate mathematical expressions is a valuable skill in everyday life. Whether calculating measurements for a home improvement project, understanding financial calculations, or simply solving problems that arise in daily situations, a strong foundation in mathematics is essential. The process of simplifying expressions like the one discussed involves breaking down complex problems into manageable steps, which is a valuable skill in any context. Furthermore, the practice of working with square roots and decimals reinforces numerical literacy and attention to detail, both of which are important in various aspects of life. By mastering the evaluation of mathematical expressions, individuals can enhance their problem-solving abilities, improve their analytical skills, and gain a deeper understanding of the world around them. From scientific research to everyday decision-making, the relevance of these skills cannot be overstated.
Conclusion
In summary, evaluating the expression 0.33 × (√50 - √18) + (√2)/2 involves a series of methodical steps, each building upon the previous one to arrive at the final solution. Our journey through this mathematical problem has highlighted several key concepts and techniques, from simplifying square roots to applying the order of operations and handling decimal approximations. We began by recognizing the need to simplify the square roots, transforming √50 into 5√2 and √18 into 3√2. This simplification was crucial in making the subsequent subtraction easier. By breaking down the numbers under the square root into their prime factors and identifying perfect squares, we were able to express the radicals in their simplest forms. Next, we addressed the subtraction within the parentheses, combining the simplified square roots to get 2√2. This step demonstrated the importance of recognizing like terms and combining them effectively. We then turned our attention to the decimal multiplication, converting 0.33 into its fractional equivalent, 1/3, for more accurate calculations. This substitution allowed us to rewrite the expression as (1/3) × (2√2), which simplified to (2√2)/3. The subsequent addition required us to find a common denominator, which was 6. By converting both fractions to have this denominator, we were able to add them and arrive at the simplified exact form of the expression: (7√2)/6. To obtain a numerical approximation, we substituted the value of √2 ≈ 1.414, leading us to an approximate value of 1.6497. We also explored alternative approaches to solving the expression, including using decimal approximations throughout the process and factoring out common terms. These alternative methods provided additional insights and demonstrated the flexibility in problem-solving strategies. Furthermore, we discussed the practical applications and relevance of such calculations in various fields, ranging from physics and engineering to finance and computer graphics. The ability to simplify and evaluate mathematical expressions is a fundamental skill that has broad implications in both academic and real-world contexts. By understanding the underlying principles and employing a step-by-step approach, we can confidently tackle complex mathematical problems and apply these skills to a wide range of situations. The process of solving this expression not only reinforces mathematical concepts but also enhances problem-solving abilities and analytical thinking, which are valuable assets in any endeavor. Thus, the evaluation of 0.33 × (√50 - √18) + (√2)/2 serves as an excellent example of how mathematical skills are essential for both theoretical understanding and practical application.
