Solve √x = 7: Step-by-Step Solution In Simplest Form

Introduction

Hey guys! Today, we're diving into a fundamental math problem: solving the equation √x = 7. This might seem straightforward, but it's crucial to understand the steps involved to ensure we arrive at the exact simplest form. Whether you're brushing up on your algebra skills or tackling homework, this guide will walk you through the process step by step. We’ll cover the basics of square roots, how to isolate the variable, and how to express your answer in its most simplified form. So, let’s get started and unravel this mathematical puzzle together!

Understanding the Basics of Square Roots

Before we jump into solving the equation, let's quickly recap what square roots are all about. The square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Mathematically, we denote the square root using the radical symbol, √. When we see √x, it means we're looking for a number that, when squared, equals x. It’s super important to remember that square roots can only deal with non-negative numbers if we're working within the realm of real numbers. This is because no real number, when multiplied by itself, will give you a negative result. Therefore, the expression under the square root (the radicand) must be greater than or equal to zero. This understanding will be vital as we solve our equation and check our solution for validity. Square roots are the inverse operation of squaring, meaning they “undo” each other. This relationship is key to solving equations involving square roots. By understanding this fundamental concept, we can manipulate equations to isolate the variable and find our solution. Keep this in mind as we move forward; it's the cornerstone of solving equations like √x = 7 and more complex problems in algebra. The world of square roots opens the door to many other mathematical concepts, so mastering this basic principle is super beneficial for your mathematical journey. Let’s keep this knowledge in our back pocket as we solve the equation!

Step-by-Step Solution to √x = 7

Alright, let’s get down to business and solve the equation √x = 7. The goal here is to isolate x on one side of the equation. Since x is currently under the square root, we need to perform an operation that will undo the square root. As we mentioned earlier, the inverse operation of taking a square root is squaring. So, to get rid of the square root, we're going to square both sides of the equation. This is a crucial step, guys, because whatever you do to one side of the equation, you absolutely must do to the other to maintain balance. Squaring both sides gives us (√x)² = 7². Now, let's simplify. On the left side, the square root and the square cancel each other out, leaving us with just x. On the right side, 7² means 7 * 7, which equals 49. So, our equation now looks like x = 49. It seems like we've found our solution, but we need to be sure. Whenever we solve equations involving square roots, it's super important to check our answer to make sure it's valid. This is because squaring both sides can sometimes introduce extraneous solutions – values that satisfy the transformed equation but not the original. To check our solution, we’ll substitute x = 49 back into the original equation: √49 = 7. The square root of 49 is indeed 7, so our solution checks out! Therefore, the solution to the equation √x = 7 is x = 49. We’ve successfully isolated x and found its value. This step-by-step process is a fundamental technique in algebra, and mastering it will help you tackle more complex equations with confidence.

Expressing the Answer in Exact Simplest Form

Now that we've found our solution, x = 49, it's crucial to express it in the exact simplest form. In this case, the solution is already a whole number, so there’s not much simplification needed. However, it's always a good practice to ensure your answer is as clean and clear as possible. The term “exact” means we should avoid approximations, like decimals, unless specifically asked for. Since 49 is a perfect square, its square root is a whole number, which is already in its most precise form. There are no fractions to reduce, no radicals to simplify further, and no common factors to eliminate. So, x = 49 is indeed in its exact simplest form. The solution set is simply {49}. Guys, sometimes, solutions might involve square roots that can't be simplified into whole numbers, such as √2 or √3. In such cases, we leave the answer in its radical form because that's the most exact representation. We avoid converting them into decimal approximations to maintain accuracy. Understanding what