Simplifying Radicals Solving $\sqrt{40}+2 \sqrt{10}+\sqrt{90}$
Are you grappling with simplifying radical expressions? This article provides a comprehensive breakdown of how to solve the expression $\sqrt{40}+2 \sqrt{10}+\sqrt{90}$. We will guide you through each step, ensuring you understand the underlying principles of simplifying radicals. By the end of this guide, you’ll be able to confidently tackle similar problems and enhance your understanding of mathematical operations involving square roots.
Understanding the Basics of Radical Simplification
Before diving into the solution, it's crucial to grasp the fundamental concepts of simplifying radicals. A radical is a mathematical expression involving a root, most commonly a square root. Simplifying radicals involves reducing the expression inside the square root to its simplest form. This often means factoring out perfect squares from the radicand (the number inside the square root symbol).
To effectively simplify radicals, remember these key principles:
- Perfect Squares: Identify perfect square factors within the radicand (e.g., 4, 9, 16, 25, etc.).
- Product Property of Radicals: The square root of a product is the product of the square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
- Combining Like Terms: You can only add or subtract radicals if they have the same radicand (the number inside the square root).
With these principles in mind, we can now approach the problem systematically.
Breaking Down $\sqrt{40}$
Our first step is to simplify $\sqrt{40}$. To do this, we need to find the largest perfect square that divides 40. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. Among these, 4 is a perfect square (2 * 2 = 4). We can rewrite $\sqrt{40}$ as follows:
Now, using the product property of radicals, we can separate the square root:
Since $\sqrt{4} = 2$, the simplified form of $\sqrt{40}$ is:
This transformation is crucial because it allows us to combine like terms later in the problem. Identifying and extracting perfect square factors is the essence of simplifying square roots. Understanding this process will help in tackling more complex expressions involving radicals.
Simplifying $\sqrt{90}$
Next, we turn our attention to $\sqrt{90}$. Similar to our approach with $\sqrt{40}$, we need to identify the largest perfect square that divides 90. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. Among these, 9 is a perfect square (3 * 3 = 9). Therefore, we can rewrite $\sqrt{90}$ as:
Applying the product property of radicals, we get:
Since $\sqrt{9} = 3$, the simplified form of $\sqrt{90}$ is:
By extracting the perfect square factor, we have successfully simplified $\sqrt{90}$ into a form that can be combined with other terms containing $\sqrt{10}$. This step highlights the importance of recognizing perfect squares and their role in simplifying radical expressions. The ability to break down complex radicals into simpler forms is a fundamental skill in algebra and is essential for solving equations involving radicals.
Combining the Simplified Terms
Now that we have simplified $\sqrt{40}$ and $\sqrt{90}$, we can substitute these simplified forms back into the original expression:
Notice that all terms now contain the same radical, $\sqrt{10}$. This allows us to combine these terms by adding their coefficients. Think of $\sqrt{10}$ as a common unit, similar to how you would combine like terms in algebraic expressions (e.g., 2x + 2x + 3x).
Adding the coefficients, we get:
Therefore, the simplified form of the original expression is:
This final step demonstrates the power of simplifying radicals. By breaking down the original radicals and then combining like terms, we arrived at a concise and manageable expression. The ability to manipulate and simplify radical expressions is crucial in various areas of mathematics, including algebra, geometry, and calculus. Understanding these techniques allows for more efficient problem-solving and a deeper comprehension of mathematical concepts.
Conclusion: Mastering Radical Simplification
In this article, we have thoroughly explored how to simplify the expression $\sqrt40}+2 \sqrt{10}+\sqrt{90}$. We began by understanding the basics of radical simplification, emphasizing the importance of identifying perfect squares and applying the product property of radicals. We then methodically simplified $\sqrt{40}$ and $\sqrt{90}$, extracting perfect square factors to rewrite them in simpler forms. Finally, we combined the simplified terms, demonstrating how like radicals can be added together to reach the final answer$.
Key takeaways from this guide include:
- Identifying perfect square factors is crucial for simplifying radicals.
- The product property of radicals allows us to separate and simplify square roots.
- Combining like terms is only possible when the radicands are the same.
By mastering these concepts, you can confidently approach and solve a wide range of problems involving radical expressions. Remember, practice is key. Work through various examples to reinforce your understanding and build your skills. With consistent effort, simplifying radicals will become second nature, enhancing your overall mathematical proficiency. Keep exploring and challenging yourself with more complex problems to further solidify your knowledge.
Therefore, the choice equivalent to the expression $\sqrt{40}+2 \sqrt{10}+\sqrt{90}$ is:
A. $7 \sqrt{10}$
Practice Problems
To further solidify your understanding of simplifying radical expressions, try solving these practice problems:
- Simplify $\sqrt{75} + 3\sqrt{3} - \sqrt{12}$
- Simplify $2\sqrt{20} - \sqrt{45} + \sqrt{80}$
- Simplify $\sqrt{24} + \sqrt{54} - \sqrt{6}$
Working through these problems will help you apply the techniques discussed in this article and build your confidence in simplifying radicals. Good luck, and happy simplifying!
