Simplifying The Expression √50 + √2 And Determining Its Nature
This article delves into the simplification and classification of the mathematical expression √50 + √2. We will explore how to simplify the expression, determine whether it is rational or irrational, and ultimately arrive at the correct answer. Understanding the nature of square roots and how they interact is crucial in mathematics, and this example provides a clear illustration of these concepts. Let's embark on a journey to unlock the secrets hidden within this seemingly simple expression.
Understanding the Basics: Rational vs. Irrational Numbers
Before we dive into the simplification process, it's essential to understand the difference between rational and irrational numbers. This distinction is key to classifying the final result of our expression.
A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. This includes integers themselves (e.g., 5 can be written as 5/1), terminating decimals (e.g., 0.25 can be written as 1/4), and repeating decimals (e.g., 0.333... can be written as 1/3). The ability to represent a number as a precise ratio of two integers is the defining characteristic of rationality.
On the other hand, an irrational number cannot be expressed as a fraction of two integers. When written as a decimal, irrational numbers neither terminate nor repeat. The most famous example of an irrational number is π (pi), which is approximately 3.14159... but continues infinitely without any repeating pattern. Another common example is the square root of a non-perfect square, such as √2 or √5. These numbers have decimal representations that go on forever without repeating, making them fundamentally different from rational numbers. Identifying whether a number is rational or irrational is a fundamental skill in mathematics, with implications for various areas, including algebra, geometry, and calculus.
Step-by-Step Simplification of √50 + √2
Now, let's tackle the expression √50 + √2 step by step. The key to simplifying this expression lies in recognizing that √50 can be further simplified. We need to find the largest perfect square that divides 50. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). In this case, 25 is the largest perfect square that divides 50 (50 = 25 * 2).
We can rewrite √50 as follows:
√50 = √(25 * 2)
Using the property of square roots that states √(a * b) = √a * √b, we can separate the square root:
√(25 * 2) = √25 * √2
Since √25 is 5, we have:
√25 * √2 = 5√2
Now we can substitute this simplified form back into the original expression:
√50 + √2 = 5√2 + √2
Here, we have two terms with the same radical, √2. We can treat √2 as a common factor and combine the coefficients:
5√2 + √2 = (5 + 1)√2
This simplifies to:
(5 + 1)√2 = 6√2
Therefore, the simplified form of the expression √50 + √2 is 6√2. This step-by-step simplification clearly demonstrates how to break down a complex expression into manageable parts, ultimately arriving at the most concise form. Understanding these techniques is crucial for success in algebra and beyond.
Classifying the Simplified Expression: 6√2
Having simplified the expression to 6√2, the next crucial step is to classify it as either rational or irrational. As we discussed earlier, this classification depends on whether the number can be expressed as a fraction of two integers.
The number 6 is clearly an integer, and therefore rational. However, √2 is the square root of a non-perfect square, which means it is an irrational number. The product of a rational number (other than zero) and an irrational number is always irrational. This is because multiplying an irrational number by a rational number simply scales the irrational part, without eliminating its non-repeating, non-terminating decimal nature.
To understand this intuitively, consider that √2 has an infinite, non-repeating decimal representation. Multiplying it by 6 simply stretches this decimal representation but does not introduce any repeating pattern or termination. Therefore, 6√2 also has an infinite, non-repeating decimal representation, making it an irrational number.
In conclusion, the simplified expression 6√2 is classified as irrational. This classification is crucial for understanding the properties of the number and its behavior in various mathematical contexts. Recognizing the irrationality of 6√2 allows us to make accurate predictions about its decimal representation and its interactions with other numbers.
Analyzing the Answer Choices
Now that we have simplified the expression √50 + √2 to 6√2 and classified it as irrational, we can confidently analyze the given answer choices and select the correct one.
Let's revisit the choices:
A. It is irrational and equal to 2√13. B. It is irrational and equal to 6√2. C. It is rational and equal to 2. D. It is rational and equal to 6√2.
We can immediately eliminate options C and D because we have determined that the expression is irrational, not rational. This leaves us with options A and B, both of which correctly identify the expression as irrational.
Option A states that the expression is equal to 2√13. To verify this, we would need to simplify 2√13. However, 13 is a prime number, meaning it has no perfect square factors other than 1. Therefore, 2√13 cannot be simplified further, and it is clearly not equal to our simplified expression of 6√2.
Option B states that the expression is equal to 6√2. This matches our simplified result exactly. Therefore, option B is the correct answer.
By systematically simplifying the expression and classifying it, we were able to confidently eliminate incorrect answer choices and identify the correct statement. This process highlights the importance of a thorough understanding of mathematical concepts and techniques when solving problems.
Conclusion: The Correct Statement
In conclusion, the expression √50 + √2 simplifies to 6√2, which is an irrational number. Therefore, the correct statement about the expression is:
B. It is irrational and equal to 6√2.
This exercise demonstrates the importance of simplifying expressions before classifying them and highlights the key differences between rational and irrational numbers. Understanding these concepts is crucial for success in mathematics, from basic algebra to advanced calculus. By breaking down the problem into smaller steps and applying the appropriate rules and properties, we were able to confidently arrive at the correct answer. This approach can be applied to a wide range of mathematical problems, empowering you to tackle even the most challenging expressions with confidence.
