Solving √(6y + 64) For Y = 4 A Step-by-Step Guide
Introduction
In this article, we will delve into solving for the variable y in a given mathematical expression. Specifically, we are tasked with finding the value of y when y equals 4 in the equation √(6y + 64). This exercise falls under the domain of algebra, where we manipulate equations to isolate variables and determine their values. Understanding how to solve such equations is a fundamental skill in mathematics, with applications spanning various fields, from science and engineering to economics and computer science. This article aims to provide a comprehensive, step-by-step guide to solving this particular equation, ensuring clarity and understanding for readers of all backgrounds. We will begin by substituting the given value of y into the equation and then proceed with the necessary algebraic steps to simplify and arrive at the final solution. By the end of this discussion, you will not only be able to solve this specific problem but also gain a deeper appreciation for the principles of algebraic manipulation.
Understanding the Equation
The equation we are dealing with is √(6y + 64). This is a radical equation, which means it involves a square root. The expression inside the square root, 6y + 64, is a linear expression in y. To solve this equation, our primary goal is to isolate y and find its value. However, in this case, we are not solving for y in the traditional sense of finding the value(s) that make the equation equal to zero or some other constant. Instead, we are asked to evaluate the expression √(6y + 64) when y is given as 4. This is a simpler task, as we only need to substitute the value of y and simplify. The square root operation, denoted by √, is the inverse of squaring a number. Therefore, when we encounter a square root, we need to find a number that, when multiplied by itself, gives us the expression inside the square root. In this context, understanding the order of operations (PEMDAS/BODMAS) is crucial. We need to perform the operations inside the parentheses (or under the radical) first, then apply the square root. This foundational knowledge will guide us as we proceed to solve the equation. Let's break down each step to ensure clarity and accuracy.
Step-by-Step Solution
Now, let's embark on the step-by-step solution to find the value of the expression √(6y + 64) when y is 4. This process involves substitution and simplification, which are core techniques in algebra. By carefully following each step, we can arrive at the correct answer while reinforcing our understanding of algebraic principles.
Step 1: Substitute the Value of y
The first step in solving the equation is to substitute the given value of y, which is 4, into the equation. This means replacing every instance of y in the expression with the number 4. The original expression is √(6y + 64). After substitution, the expression becomes √(6 * 4 + 64). This step is crucial as it transforms the algebraic expression into an arithmetic one, which we can then simplify using the order of operations.
Step 2: Simplify the Expression Inside the Square Root
Next, we need to simplify the expression inside the square root. According to the order of operations (PEMDAS/BODMAS), we perform multiplication before addition. So, we first multiply 6 by 4, which gives us 24. The expression inside the square root now becomes 24 + 64. Then, we add 24 and 64, which results in 88. So, the expression under the square root simplifies to 88. The equation now looks like √88.
Step 3: Simplify the Square Root (if possible)
The final step is to simplify the square root, if possible. We need to determine if 88 has any perfect square factors. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). To find perfect square factors, we can prime factorize 88. The prime factorization of 88 is 2 * 2 * 2 * 11, which can be written as 2² * 2 * 11. We see that 2² (which is 4) is a perfect square factor of 88. Therefore, we can rewrite √88 as √(4 * 22). Using the property of square roots that √(a * b) = √a * √b, we can further simplify this as √4 * √22. Since √4 is 2, the simplified expression is 2√22. Thus, the final value of the expression √(6y + 64) when y is 4 is 2√22.
Detailed Explanation of Each Step
To ensure complete understanding, let's delve deeper into each step of the solution process. Breaking down each step into smaller components helps clarify the logic and mathematical principles involved. This detailed explanation will not only solidify your understanding of this specific problem but also enhance your ability to tackle similar algebraic challenges.
Detailed Explanation of Step 1: Substituting y = 4
Substitution is a fundamental concept in algebra where we replace a variable with its given value. In our case, we are substituting y with 4 in the expression √(6y + 64). The process involves identifying each occurrence of y in the expression and replacing it with the number 4. This might seem straightforward, but it's crucial to be precise to avoid errors. For instance, if the expression were more complex, such as √(6y² + 64y), substituting y with 4 would require squaring 4 in the first term and multiplying 64 by 4 in the second term. In our simpler case, we replace y with 4 in 6y, which becomes 6 * 4. The entire expression then transforms from √(6y + 64) to √(6 * 4 + 64). This substitution sets the stage for simplifying the expression using arithmetic operations.
Detailed Explanation of Step 2: Simplifying Inside the Square Root
Once we have substituted y with 4, the expression inside the square root is 6 * 4 + 64. To simplify this, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These rules dictate the sequence in which operations should be performed. In our expression, we have multiplication and addition. According to PEMDAS/BODMAS, multiplication comes before addition. Therefore, we first multiply 6 by 4, which yields 24. The expression then becomes 24 + 64. Now, we perform the addition: 24 plus 64 equals 88. Thus, the expression inside the square root simplifies to 88. This step is crucial because it reduces the complexity of the expression under the radical, making it easier to handle in the next step.
Detailed Explanation of Step 3: Simplifying the Square Root
The final step involves simplifying the square root of 88, √88. Simplifying a square root means expressing it in its simplest form, which typically involves extracting any perfect square factors. A perfect square is a number that is the square of an integer (e.g., 1, 4, 9, 16, 25, etc.). To find perfect square factors of 88, we can perform prime factorization. Prime factorization is the process of breaking down a number into its prime factors, which are numbers divisible only by 1 and themselves. The prime factorization of 88 is 2 * 2 * 2 * 11. We can rewrite this as 2² * 2 * 11. Here, 2², which equals 4, is a perfect square. This allows us to rewrite √88 as √(4 * 22). Using the property of square roots that √(a * b) = √a * √b, we can separate the square root as √4 * √22. The square root of 4 is 2, so we have 2√22. Since 22 has no further perfect square factors, this is the simplest form of the square root. Therefore, the final value of the expression √(6y + 64) when y is 4 is 2√22. This step demonstrates the importance of understanding prime factorization and the properties of square roots in simplifying mathematical expressions.
Alternative Methods to Solve
While the step-by-step method we've outlined is a direct and effective way to solve the equation, it's always beneficial to explore alternative approaches. Different methods can provide a deeper understanding of the problem and enhance problem-solving skills. In this section, we will discuss an alternative method to solve for the value of √(6y + 64) when y equals 4. This alternative approach will reinforce the concepts we've already covered and offer a fresh perspective on the problem.
Alternative Method: Direct Substitution and Simplification
An alternative method to solve this problem involves a more direct approach to substitution and simplification, focusing on efficiency and a streamlined process. Instead of breaking down the simplification into multiple steps, we can combine them to reach the solution more quickly. This method is particularly useful for those who are comfortable with the order of operations and can perform calculations mentally or with minimal intermediate steps. The key is to substitute the value of y directly into the expression and then simplify the resulting arithmetic expression without explicitly stating each intermediate result. Let's walk through this method step-by-step.
Step 1: Direct Substitution
As before, we begin by substituting y with 4 in the expression √(6y + 64). This gives us √(6 * 4 + 64). The emphasis here is on performing this substitution mentally, without writing it down as a separate step. This requires a clear understanding of what the substitution entails and how it transforms the expression.
Step 2: Combined Simplification
In this step, we combine the simplification operations. First, we perform the multiplication: 6 * 4 = 24. Then, we add this result to 64: 24 + 64 = 88. This combined calculation leads us to √88. The efficiency of this step comes from performing the arithmetic operations in sequence without pausing to write down intermediate results. This approach requires a good grasp of mental arithmetic and the order of operations.
Step 3: Simplified Square Root
Finally, we simplify the square root of 88, √88. As we discussed earlier, 88 can be factored into 4 * 22, where 4 is a perfect square. Therefore, √88 can be written as √(4 * 22), which simplifies to √4 * √22. Since √4 is 2, the simplified expression is 2√22. This final simplification step is identical to the one in our step-by-step method, but the key difference is the streamlined approach to the initial calculations. By combining the multiplication and addition steps, we arrive at the simplified square root more quickly.
Conclusion
In conclusion, we have successfully solved for the value of the expression √(6y + 64) when y equals 4. Through a detailed step-by-step approach, we first substituted the value of y, then simplified the expression inside the square root, and finally, simplified the square root itself. This process not only provided us with the solution, 2√22, but also reinforced fundamental algebraic principles such as substitution, order of operations, and simplification of radicals. Furthermore, we explored an alternative method that combined the simplification steps, demonstrating that there can be multiple pathways to arrive at the same correct answer. Understanding these different methods can enhance problem-solving flexibility and efficiency.
This exercise exemplifies the importance of a systematic approach to mathematical problems. By breaking down complex expressions into manageable steps, we can tackle even challenging equations with confidence. The skills and concepts discussed in this article are foundational to more advanced mathematical topics, making it crucial to master them. Whether you are a student learning algebra for the first time or someone looking to refresh your math skills, the principles outlined here will serve as valuable tools in your mathematical journey. Remember, practice is key to proficiency, so continue to apply these methods to various problems to solidify your understanding and build your problem-solving abilities.
