Evaluating √121 - √100 A Step By Step Solution

Hey guys! Let's dive into a fun math problem today. We're going to evaluate the expression √121 - √100, and don't worry, it's not as scary as it looks! We'll break it down step by step, making sure everyone understands each part of the process. So, grab your thinking caps, and let's get started!

Understanding Square Roots

Before we jump into the problem, 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. Think of it like this: if you have a square garden with an area of 25 square feet, the length of one side is the square root of 25, which is 5 feet (because 5 * 5 = 25). So, when we see √ symbol, we're asking ourselves, "What number times itself equals the number under the root?"

Square roots are a fundamental concept in mathematics, and understanding them is crucial for various calculations and problem-solving scenarios. A strong grasp of square roots not only helps in simplifying expressions but also in solving equations, understanding geometric concepts, and even in more advanced topics like calculus. For example, in geometry, the Pythagorean theorem, which relates the sides of a right triangle, heavily relies on the concept of square roots. Similarly, in physics, understanding square roots is essential when dealing with concepts like velocity and acceleration. Therefore, mastering square roots is an investment in your mathematical journey, opening doors to a deeper understanding of various scientific and mathematical principles.

When tackling expressions involving square roots, it's essential to remember a few key properties. First, the square root of a positive number always yields two solutions: a positive root and a negative root. However, in most basic arithmetic problems, we focus on the principal, or positive, square root. For example, while both 10 and -10 are square roots of 100, we typically consider 10 as the principal square root. Second, the square root of a product is equal to the product of the square roots. This property can be incredibly useful when simplifying complex expressions. For instance, √(4 * 9) can be simplified as √4 * √9, which equals 2 * 3, or 6. Lastly, it's important to note that the square root of a negative number is not a real number. This is because no real number, when multiplied by itself, can result in a negative number. This concept introduces us to the realm of imaginary numbers, a fascinating area in mathematics that extends beyond the real number system.

Evaluating √121

Okay, let's start with the first part of our expression: √121. We need to figure out what number, when multiplied by itself, equals 121. Think of your multiplication tables! Do you remember any number that, when squared, gives you 121? If you guessed 11, you're absolutely right! Because 11 * 11 = 121, the square root of 121 is 11. So, we can rewrite √121 as simply 11. This step is all about recognizing perfect squares – numbers that are the result of squaring a whole number. Being familiar with perfect squares (like 4, 9, 16, 25, etc.) will make evaluating square roots much easier and faster.

Recognizing perfect squares is a crucial skill when simplifying square roots, and it significantly speeds up the evaluation process. Perfect squares are numbers that result from squaring an integer (a whole number). Some common examples include 1 (1^2), 4 (2^2), 9 (3^2), 16 (4^2), 25 (5^2), and so on. The more familiar you are with these numbers, the quicker you can identify and simplify square roots. For instance, when you see √81, immediately recognizing that 81 is a perfect square (9^2) allows you to simplify it to 9 without further calculations. This skill is particularly useful when dealing with larger numbers or more complex expressions. Memorizing the first few perfect squares and their corresponding square roots can save you a lot of time and effort in problem-solving.

When dealing with square roots of larger numbers, there are a few strategies you can employ to determine if they are perfect squares. One approach is to try dividing the number by consecutive integers, starting from 2, to see if you find a factor that, when multiplied by itself, equals the original number. For example, if you are trying to find the square root of 144, you might start by dividing it by 2, then 3, and so on, until you find a factor that works (in this case, 12). Another useful technique is to consider the last digit of the number. Perfect squares have specific patterns in their last digits. For instance, a perfect square can only end in 0, 1, 4, 5, 6, or 9. So, if you encounter a number ending in 2, 3, 7, or 8, you can immediately conclude that it is not a perfect square. By combining these strategies with your knowledge of perfect squares, you can confidently tackle a wide range of square root problems.

Evaluating √100

Next up, we need to tackle √100. Just like before, we're looking for a number that, when multiplied by itself, equals 100. Can you think of it? You got it – it's 10! Because 10 * 10 = 100, the square root of 100 is 10. So, we can replace √100 with 10. This is another common square root that's helpful to memorize. Recognizing these simple square roots will make solving expressions like this much smoother.

Memorizing common square roots is an invaluable strategy for simplifying mathematical expressions and solving problems more efficiently. Certain numbers appear frequently in mathematical calculations, and knowing their square roots by heart can save you considerable time and effort. For example, knowing that √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, and √100 = 10 allows you to quickly simplify expressions without resorting to calculators or lengthy calculations. These square roots are the building blocks for many mathematical problems, and familiarity with them will significantly enhance your problem-solving skills. Think of it as having a mental toolkit of common mathematical facts readily available whenever you need them.

Beyond the basic square roots, it's also beneficial to recognize the square roots of multiples of 100. These numbers often appear in real-world applications and more complex mathematical problems. For instance, knowing that √400 = 20, √900 = 30, √1600 = 40, and so on, can help you quickly estimate the magnitudes of larger square roots and simplify calculations involving large numbers. These square roots are essentially extensions of the basic square roots, and they follow a simple pattern that makes them easy to memorize. By expanding your knowledge of square roots to include these multiples of 100, you'll be well-equipped to tackle a broader range of mathematical challenges. This knowledge is particularly useful in fields like physics and engineering, where calculations often involve large numbers and approximations.

Putting It All Together

Now that we've figured out the square roots individually, let's put them back into our original expression. We started with √121 - √100. We found that √121 = 11 and √100 = 10. So, we can rewrite the expression as 11 - 10. This is a simple subtraction problem, right? 11 minus 10 equals 1. So, the final answer is 1!

The final step in evaluating expressions often involves combining the simplified terms through basic arithmetic operations like addition, subtraction, multiplication, or division. This step is crucial for arriving at the final answer and requires careful attention to the order of operations (PEMDAS/BODMAS). In our example, after simplifying √121 to 11 and √100 to 10, we were left with the simple subtraction problem 11 - 10. However, in more complex expressions, you might encounter a mix of operations that need to be performed in the correct sequence. For instance, you might have an expression that involves both multiplication and addition, or parentheses and exponents. In such cases, following the order of operations ensures that you arrive at the correct result.

Attention to detail is paramount when performing the final calculations. Even a small error in addition, subtraction, multiplication, or division can lead to a completely different answer. It's always a good practice to double-check your calculations, especially in situations where accuracy is critical. One common mistake is overlooking negative signs or misapplying the distributive property. Another frequent error is failing to correctly simplify fractions or exponents. To minimize these errors, it's helpful to write out each step of your calculation clearly and systematically. This allows you to easily review your work and identify any potential mistakes. Additionally, using a calculator can help reduce the risk of arithmetic errors, but it's still essential to understand the underlying mathematical principles and verify that the calculator's output makes sense in the context of the problem.

The Answer

So, after evaluating the expression √121 - √100, we found that the answer is 1. And yes, the expression is a real number! We successfully simplified the square roots and performed the subtraction to arrive at our final answer. Great job, guys! You've tackled this math problem like pros. Keep practicing, and you'll become even more confident in your math skills!

A. The expression is a real number. 1