Simplify $2 \sqrt{80} + \sqrt{125} - 3 \sqrt{20}$ A Step-by-Step Guide

In the realm of mathematics, simplifying radical expressions is a fundamental skill. This article aims to provide a comprehensive guide on how to simplify the expression $2 \sqrt{80} + \sqrt{125} - 3 \sqrt{20}$. We'll break down each step, providing clear explanations and insights to enhance your understanding.

Understanding Radicals

To effectively simplify radical expressions, a solid grasp of what radicals are is crucial. At its core, a radical, often denoted by the symbol $\sqrt{}$, represents the root of a number. The most common type is the square root, which asks: “What number, when multiplied by itself, equals the number under the radical?” For instance, $\sqrt{9} = 3$ because 3 * 3 = 9. However, the numbers under the radical, known as radicands, are not always perfect squares. This is where simplification becomes necessary. Simplification involves breaking down the radicand into its prime factors and extracting any perfect square factors. By understanding this fundamental concept, we can approach more complex expressions with confidence and precision. For example, $\sqrt{8}$ can be simplified because 8 can be factored into 4 * 2, and 4 is a perfect square. This understanding forms the bedrock for tackling expressions such as the one we aim to simplify: $2 \sqrt{80} + \sqrt{125} - 3 \sqrt{20}$. Without a clear understanding of radicals, the process of simplification can seem daunting, but with a firm grasp of the basics, it becomes a methodical and achievable task. The ability to identify and extract perfect square factors is the key to unlocking simplification, making it an essential skill in algebra and beyond.

Step 1: Simplifying $\sqrt{80}$

The initial step in simplifying our expression involves tackling the first radical term: $2 \sqrt{80}$. To simplify $\sqrt{80}$, we need to identify the largest perfect square that divides 80. Let's break down 80 into its prime factors: 80 = 2 * 2 * 2 * 2 * 5 = $2^4$ * 5. We can see that $2^4$ (which is 16) is a perfect square. Thus, we can rewrite $\sqrt{80}$ as $\sqrt{16 * 5}$. Using the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we can further simplify this to $\sqrt{16} * \sqrt{5}$. Since $\sqrt{16}$ is 4, we now have $4 \sqrt{5}$. Remember that we originally had $2 \sqrt{80}$, so we multiply our simplified radical by 2, resulting in 2 * $4 \sqrt{5}$ = $8 \sqrt{5}$. This meticulous breakdown allows us to transform a seemingly complex radical into a more manageable form. The ability to identify perfect square factors within a radicand is a critical skill in simplifying radicals. By applying this technique, we've successfully simplified the first term of our expression, setting the stage for further simplification. This step-by-step approach not only clarifies the process but also reduces the chances of errors, ensuring accuracy in our calculations. The simplification of $\sqrt{80}$ to $4 \sqrt{5}$ is a crucial building block in solving the overall expression.

Step 2: Simplifying $\sqrt{125}$

Next, we focus on simplifying the second term in our expression: $\sqrt{125}$. Similar to the previous step, our goal is to identify the largest perfect square that divides 125. To do this, we can factorize 125 into its prime factors: 125 = 5 * 5 * 5 = $5^3$. We observe that $5^2$ (which is 25) is a perfect square. Consequently, we can rewrite $\sqrt{125}$ as $\sqrt{25 * 5}$. Applying the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we simplify this to $\sqrt{25} * \sqrt{5}$. Since $\sqrt{25}$ equals 5, we are left with $5 \sqrt{5}$. This simplification effectively transforms $\sqrt{125}$ into a more concise and manageable form. The process of identifying perfect square factors within the radicand is a cornerstone of simplifying radicals. By breaking down 125 into its prime factors and recognizing the perfect square of 25, we were able to extract it from the radical. This methodical approach not only simplifies the radical expression but also enhances our understanding of the underlying mathematical principles. The simplification of $\sqrt{125}$ to $5 \sqrt{5}$ is another significant step in solving the overall expression, bringing us closer to the final simplified form. This step reinforces the importance of prime factorization and the identification of perfect squares in simplifying radicals.

Step 3: Simplifying $3 \sqrt{20}$

Now, let's simplify the third term: $3 \sqrt{20}$. Our strategy remains consistent: find the largest perfect square that divides 20. Factoring 20 into its prime factors gives us 20 = 2 * 2 * 5 = $2^2$ * 5. Here, we see that $2^2$ (which is 4) is a perfect square. We can rewrite $\sqrt{20}$ as $\sqrt{4 * 5}$. Applying the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we get $\sqrt{4} * \sqrt{5}$. Since $\sqrt{4}$ is 2, we have $2 \sqrt{5}$. However, we must remember the original coefficient of 3 in front of the radical, so we multiply our simplified radical by 3, giving us 3 * $2 \sqrt{5}$ = $6 \sqrt{5}$. This careful step ensures that we account for all parts of the original expression. The ability to identify and extract perfect square factors is crucial in simplifying radicals, and this step further illustrates that principle. By breaking down 20 into its prime factors and recognizing the perfect square of 4, we were able to simplify the radical effectively. The multiplication by the original coefficient of 3 highlights the importance of maintaining accuracy throughout the simplification process. The result, $6 \sqrt{5}$, is a simplified form of $3 \sqrt{20}$ and is a key component in solving the overall expression.

Step 4: Combining Like Terms

After simplifying each radical term individually, we can now combine like terms. Our expression has been transformed into $8 \sqrt{5} + 5 \sqrt{5} - 6 \sqrt{5}$. Notice that all three terms contain the same radical, $\sqrt{5}$. This allows us to combine the coefficients in front of the radical. Think of $\sqrt{5}$ as a common variable, like 'x'. So, we have 8x + 5x - 6x. Combining the coefficients, we perform the arithmetic: 8 + 5 - 6 = 7. Therefore, our simplified expression is $7 \sqrt{5}$. This final step demonstrates the power of simplifying radicals – by reducing each term to its simplest form, we can then easily combine them. The ability to recognize and combine like terms is a fundamental skill in algebra, and it is crucial for simplifying radical expressions. By treating the radical as a common variable, we can apply basic arithmetic operations to the coefficients, leading to a simplified final answer. The combination of like terms is the culmination of the simplification process, bringing together the results of the previous steps into a concise and elegant solution. The final simplified form, $7 \sqrt{5}$, represents the original expression in its most basic and understandable form.

Final Answer

Therefore, the simplified form of the expression $2 \sqrt{80} + \sqrt{125} - 3 \sqrt{20}$ is $7 \sqrt{5}$. This result showcases the effectiveness of breaking down complex radical expressions into simpler components and then combining like terms. The step-by-step approach, involving prime factorization, identification of perfect square factors, and careful arithmetic, is essential for accurate simplification. The final answer, $7 \sqrt{5}$, not only represents the simplified form but also demonstrates a clear understanding of radical simplification techniques. This comprehensive process provides a solid foundation for tackling more complex algebraic problems involving radicals. The journey from the initial expression to the final simplified form highlights the importance of methodical problem-solving and attention to detail in mathematics. The ability to simplify radical expressions is a valuable skill, and the result, $7 \sqrt{5}$, is a testament to the power of this skill.