Calculating Absolute Value Of Square Roots: A Step-by-Step Guide

Hey guys, let's dive into a cool math problem! We're gonna figure out the value of |√(123 - 456) + √(123 + 456)|. Sounds a bit intimidating, right? But trust me, we'll break it down step by step, and it'll be a piece of cake. This problem involves square roots and absolute values, so let's get our math hats on and get started! The key here is to understand the absolute value and how it affects the result of our calculation, especially when dealing with square roots of negative numbers. Don't worry, we'll cover it all.

Understanding the Basics: Absolute Value and Square Roots

Alright, before we jump into the main problem, let's quickly recap what absolute value and square roots are all about.

Firstly, absolute value, denoted by |x|, is the distance of a number 'x' from zero on the number line. It's always a non-negative value. In simple terms, if x is positive or zero, then |x| = x. But if x is negative, then |x| = -x. Think of it like this: it turns any negative number into its positive counterpart. For example, |-5| = 5, and |5| = 5. Absolute value is super important because it ensures that our final answer is never negative, regardless of what's inside those bars. This function is an essential concept in various areas of mathematics, from basic arithmetic to advanced calculus. Understanding it will help solve many types of problems.

Secondly, square roots. 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 * 3 = 9. We denote the square root using the symbol √. Now, a little twist: the square root of a negative number is not a real number. If you try to calculate the square root of a negative number using real numbers, you will get an error. This is where complex numbers come into play, but for this problem, we will make sure our final answer is real and positive, thanks to the use of absolute value.

Now we've got the basics covered, let's move on to the actual calculation. Keep these concepts in mind, as they're gonna be crucial as we solve this problem!

Step-by-Step Calculation: Breaking Down the Problem

Alright, time to get our hands dirty and start solving the problem. The expression we need to calculate is |√(123 - 456) + √(123 + 456)|. Let’s break it down into smaller, manageable steps.

First, let's deal with the inner parts of the square roots. We have two expressions: (123 - 456) and (123 + 456). Let's calculate them separately:

• 123 - 456 = -333
• 123 + 456 = 579

Now, we can rewrite our expression as |√(-333) + √(579)|. Here, we can see that we have a problem: the square root of -333 is not a real number. But don't you worry, because the absolute value can save us. To deal with this, we should consider that √(-333) = √(333 * -1) = √(333) * √(-1). √(-1) is represented by the imaginary unit 'i'. Since we know that √(-1) results in an imaginary number, we must consider this when we are calculating the value in the problem. If we continue with our real number calculations, it could lead to incorrect results.

However, in this context, the absolute value function implies that we only need to consider the magnitude of the final result. In this case, since the expression contains an imaginary number, the result will also be imaginary. Since the absolute value of any complex number, a + bi, is √(a^2 + b^2), we would need to account for the real and imaginary parts. But let's simplify our approach and recognize that the presence of the square root of a negative number indicates that we have an imaginary component. The absolute value then simply refers to the magnitude of the complex number. So, in this specific case, the absolute value helps us focus on the overall size or magnitude of the final complex number without the detailed step-by-step of the calculation.

Next, the expression now is |√(-333) + √(579)|. The √(579) is a positive real number. You can find the approximate value using a calculator, √(579) ≈ 24.06. And √(-333) = √333 * √-1, or approximately 18.25i. Therefore, √(-333) + √(579) ≈ 18.25i + 24.06. Since the absolute value gives us the magnitude, |18.25i + 24.06| ≈ 30.07. So, the absolute value converts the complex number into a positive real number indicating its magnitude.

Important Considerations and Potential Pitfalls

Let’s discuss some key points and potential pitfalls you might encounter while solving similar problems. Recognizing these can help you avoid common mistakes and solve problems more effectively.

One common mistake is forgetting the order of operations, often remembered by the acronym PEMDAS or BODMAS. This stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Always perform calculations in this order. Secondly, you need to understand that the square root of a negative number is not a real number. This is a crucial detail, as it changes the nature of the solution. You must remember that absolute values deal with distances from zero. Always be careful about signs! A small mistake in adding or subtracting can throw off the entire answer.

In this specific problem, the key is to recognize that we are dealing with a complex number due to the square root of -333. Although the absolute value gives us the magnitude, the presence of an imaginary component indicates this. If you are ever unsure, it’s always a good idea to double-check your calculations, especially with the use of a calculator. Understanding the basics of absolute values, square roots, and complex numbers will make these types of problems much easier. Always remember to break down complex problems into simpler steps.

Conclusion: Wrapping It Up

So, after breaking it down step by step, we found that |√(123 - 456) + √(123 + 456)| is approximately 30.07. This problem combines different math concepts, including square roots, absolute values, and also introduces the concept of complex numbers. The presence of the square root of a negative number gave us a complex number, and we used the absolute value function to get the magnitude of the resulting complex number. It's a great example of how different concepts in mathematics are interconnected. I hope this explanation has been helpful. Keep practicing and exploring new problems, and your math skills will continue to improve! Remember, the key is to break down the problem into smaller, manageable steps, and always double-check your work. Happy calculating, guys!