Simplify Radical Expressions: A Step-by-Step Guide To Solving 11√45 - 4√5

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. This article delves into the process of finding the difference between 11√45 and 4√5, providing a step-by-step solution and explanation to enhance your understanding of radical simplification. We will explore the underlying concepts and techniques, ensuring you can confidently tackle similar problems in the future.

Understanding Radicals: The Building Blocks

Before we embark on the solution, it's crucial to grasp the essence of radicals. A radical, denoted by the symbol √, represents the root of a number. For instance, √9 signifies the square root of 9, which is 3, as 3 multiplied by itself equals 9. Radicals can also represent cube roots, fourth roots, and so on. The number under the radical symbol is termed the radicand.

Simplifying radicals involves expressing them in their simplest form, where the radicand has no perfect square factors (for square roots), perfect cube factors (for cube roots), and so forth. This simplification process often makes it easier to perform operations such as addition, subtraction, multiplication, and division involving radicals.

Deconstructing the Problem: 11√45 - 4√5

Our mission is to determine the difference between 11√45 and 4√5. To achieve this, we must first simplify the radicals involved. Let's begin by simplifying √45. We need to identify the largest perfect square that divides 45. The perfect squares less than 45 are 1, 4, 9, 16, 25, and 36. Among these, 9 is the largest perfect square that divides 45 (45 = 9 × 5).

Thus, we can rewrite √45 as √(9 × 5). Using the property of radicals that √(a × b) = √a × √b, we have √(9 × 5) = √9 × √5. Since √9 = 3, we can simplify √45 to 3√5. Now, substituting this back into our original expression, 11√45 becomes 11 × 3√5, which simplifies to 33√5.

Our expression now transforms into 33√5 - 4√5. We have successfully simplified the first term, and both terms now involve the same radical, √5. This sets the stage for the final step: combining like terms.

Combining Like Terms: The Final Showdown

In algebra, like terms are terms that have the same variable raised to the same power. In our case, both 33√5 and 4√5 contain the same radical, √5, making them like terms. To combine like terms, we simply add or subtract their coefficients. The coefficient is the numerical factor that multiplies the radical.

In the expression 33√5 - 4√5, the coefficients are 33 and -4. Subtracting the coefficients, we get 33 - 4 = 29. Therefore, 33√5 - 4√5 simplifies to 29√5. This is the final answer, representing the difference between 11√45 and 4√5 in its simplest form.

The Answer and Its Significance

We have successfully determined that the difference between 11√45 and 4√5 is 29√5. This corresponds to option A in the given choices. The process we followed highlights the importance of simplifying radicals before performing operations. By simplifying √45 to 3√5, we transformed the problem into one involving like terms, making the subtraction straightforward.

Understanding how to simplify radicals and combine like terms is crucial for success in algebra and beyond. It's a skill that crops up in various mathematical contexts, from solving equations to working with geometric figures. By mastering these techniques, you'll gain a deeper appreciation for the elegance and power of mathematical simplification.

Expanding Our Understanding: Additional Insights

To further solidify your understanding, let's explore some related concepts and techniques. We encountered the property √(a × b) = √a × √b during our simplification process. This property is a cornerstone of radical manipulation. It allows us to break down radicals into simpler components, making them easier to work with.

Another important concept is the idea of rationalizing the denominator. This technique is used to eliminate radicals from the denominator of a fraction. For example, if we have the expression 1/√2, we can rationalize the denominator by multiplying both the numerator and denominator by √2. This gives us (1 × √2) / (√2 × √2) = √2 / 2, where the denominator is now a rational number.

Furthermore, understanding the different types of radicals, such as square roots, cube roots, and higher-order roots, is essential. Each type of radical has its own set of rules and properties. For instance, the cube root of a number is the value that, when multiplied by itself three times, equals the original number. Similarly, the fourth root is the value that, when multiplied by itself four times, equals the original number.

Practice Makes Perfect: Sharpening Your Skills

As with any mathematical concept, practice is key to mastery. To reinforce your understanding of radical simplification, try tackling similar problems. Here are a few examples to get you started:

1. Simplify 5√18 - 2√8.
2. Simplify √27 + 4√3.
3. Simplify (√20 × √15) / √3.

By working through these problems, you'll develop your skills in identifying perfect square factors, applying the properties of radicals, and combining like terms. The more you practice, the more comfortable and confident you'll become in simplifying radical expressions.

Conclusion: A Journey Through Radical Simplification

In this article, we embarked on a journey to unravel the difference between 11√45 and 4√5. We began by understanding the fundamental concepts of radicals and simplification. We then deconstructed the problem, simplified the radicals, combined like terms, and arrived at the solution: 29√5. We further expanded our understanding by exploring related concepts, such as rationalizing the denominator and working with different types of radicals.

By mastering the techniques presented in this article, you'll not only be able to solve similar problems but also gain a deeper appreciation for the beauty and logic of mathematics. Remember, practice is the key to success. So, keep exploring, keep simplifying, and keep expanding your mathematical horizons.

What is the difference between $11 oot{45} - 4 oot{5}$? What is the simplified form of $11 oot{45} - 4 oot{5}$?

Simplify Radical Expressions A Step-by-Step Guide to Solving 11√45 - 4√5