Simplifying $11 \sqrt{45} - 4 \sqrt{5}$ A Step-by-Step Guide

In the realm of mathematics, simplifying radical expressions is a fundamental skill, particularly when dealing with numbers that aren't perfect squares. In this comprehensive guide, we'll delve into simplifying the expression $11 \sqrt{45} - 4 \sqrt{5}$ by breaking it down step by step. Our focus is to simplify radicals, understand the properties involved, and apply these to solve the problem effectively. By the end of this exploration, you'll not only grasp the solution but also gain a deeper insight into handling similar radical expressions. The key to simplifying radical expressions lies in identifying perfect square factors within the radicand (the number inside the square root). This process allows us to extract these perfect squares, thereby reducing the overall expression to its simplest form. Before diving into the specifics of our problem, let's briefly review the core principles of simplifying radicals. A radical is considered simplified when the radicand has no perfect square factors other than 1, no fractions under the radical sign, and no radicals in the denominator of a fraction. The properties of radicals, such as $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, are essential tools in this simplification process. Mastering these properties will enable you to manipulate and simplify a wide range of radical expressions with confidence. Now, let's embark on our journey to simplify the given expression, uncovering the techniques and strategies involved in the process.

Breaking Down the Expression: $11 \sqrt{45} - 4 \sqrt{5}$

To effectively simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$, our primary focus should be on breaking down the radicals into their simplest form. This involves identifying and extracting any perfect square factors from the numbers inside the square roots. Let’s start by examining the first term, $11 \sqrt{45}$. Here, the radicand is 45, which can be factored into $9 \times 5$. Notice that 9 is a perfect square ($3^2$), making this factorization ideal for simplification. We can rewrite $\sqrt{45}$ as $\sqrt{9 \times 5}$. Using the property $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, we can further break this down into $\sqrt{9} \cdot \sqrt{5}$. Since $\sqrt{9}$ is 3, we now have $3 \sqrt{5}$. Multiplying this by the coefficient 11, we get $11 \cdot 3 \sqrt{5} = 33 \sqrt{5}$. Now, let’s turn our attention to the second term, $4 \sqrt{5}$. In this case, the radicand is 5, which is a prime number and has no perfect square factors other than 1. Therefore, this term is already in its simplest form and doesn't require further simplification. With both terms now examined, we can proceed to combine them. We've simplified $11 \sqrt{45}$ to $33 \sqrt{5}$, and the second term remains as $4 \sqrt{5}$. This step of breaking down the radicals is crucial in simplifying the overall expression, allowing us to identify like terms that can be combined. By focusing on extracting perfect square factors, we pave the way for the next stage of simplifying the radical expression. The initial breakdown sets the stage for a smoother and more accurate solution.

Combining Like Terms: $33 \sqrt{5} - 4 \sqrt{5}$

Having simplified the expression $11 \sqrt{45} - 4 \sqrt{5}$ to $33 \sqrt{5} - 4 \sqrt{5}$, we are now presented with a straightforward task: combining like terms. In the context of radical expressions, “like terms” refer to terms that have the same radicand (the number under the radical symbol). In this case, both terms have the same radicand, which is $\sqrt{5}$. This allows us to combine these terms in a manner similar to combining algebraic terms. The process of combining like terms involves treating the radical part ($\sqrt{5}$ in this case) as a common factor. We can then focus on the coefficients, which are the numbers multiplying the radical. In our expression, the coefficients are 33 and -4. To combine these terms, we simply perform the arithmetic operation on the coefficients: $33 - 4$. This gives us 29. Therefore, when we combine the terms $33 \sqrt{5}$ and $-4 \sqrt{5}$, we get $29 \sqrt{5}$. This step is a crucial aspect of simplifying radical expressions, as it consolidates multiple terms into a single, more manageable term. By identifying and combining like terms, we reduce the complexity of the expression, bringing us closer to the final simplified form. The ability to recognize and combine like terms is a fundamental skill in algebra and is particularly useful when dealing with radicals, polynomials, and other complex expressions. With this step completed, we have effectively simplified the original expression to its most concise form.

The Final Simplified Form: $29 \sqrt{5}$

After meticulously breaking down and combining terms, we arrive at the final simplified form of the expression $11 \sqrt{45} - 4 \sqrt{5}$, which is $29 \sqrt{5}$. This result represents the expression in its most reduced and simplest form. No further simplification is possible because the radicand, 5, is a prime number and has no perfect square factors other than 1. The coefficient, 29, is an integer, and there are no remaining like terms to combine. In summary, our journey to simplify the given expression involved several key steps. First, we identified and extracted perfect square factors from the radicand of the first term, $11 \sqrt{45}$, transforming it into $33 \sqrt{5}$. This step is crucial because it allowed us to rewrite the expression in terms of the same radical, $\sqrt{5}$. Next, we recognized that the second term, $4 \sqrt{5}$, was already in its simplest form. With both terms expressed in terms of $\sqrt{5}$, we were able to combine them as like terms. This involved subtracting the coefficients (33 and 4), which resulted in 29. Finally, we expressed the simplified expression as $29 \sqrt{5}$, which is the most concise and clear representation of the original problem. This process demonstrates the importance of understanding radical properties and the technique of simplifying radical expressions. The final answer, $29 \sqrt{5}$, not only solves the problem but also provides a clear and understandable result that highlights the beauty and efficiency of mathematical simplification.

Conclusion: Mastering Radical Simplification

In conclusion, simplifying radical expressions like $11 \sqrt{45} - 4 \sqrt{5}$ is a fundamental skill in mathematics that combines several key concepts. Through our step-by-step approach, we've demonstrated how to break down complex expressions into manageable components, making them easier to understand and solve. The process began with identifying and extracting perfect square factors from the radicand, which is a critical step in simplifying radicals. This technique allowed us to transform $11 \sqrt{45}$ into $33 \sqrt{5}$, bringing the expression closer to its simplest form. Recognizing like terms and combining them is another essential skill highlighted in this simplification process. By understanding that $33 \sqrt{5}$ and $4 \sqrt{5}$ are like terms, we were able to perform a straightforward subtraction, leading us to $29 \sqrt{5}$. This final simplified form showcases the power of mathematical simplification – taking a seemingly complex expression and reducing it to its most basic and understandable form. The ability to simplify radicals is not just a mathematical exercise; it's a crucial skill that underpins many areas of mathematics and science. From algebra to calculus, and even in physics and engineering, the ability to manipulate and simplify expressions is invaluable. By mastering these techniques, you gain a deeper understanding of mathematical principles and enhance your problem-solving abilities. As you continue your mathematical journey, remember that practice is key. The more you work with radical expressions and other mathematical concepts, the more confident and proficient you will become. Embrace challenges, explore different approaches, and never stop learning. Mathematics is a journey of discovery, and every problem solved is a step forward on that path. The simplified form $29 \sqrt{5}$ is not just an answer; it's a testament to the power of mathematical reasoning and the beauty of simplicity.