Simplify Radicals: Step-by-Step Solution

Hey guys! Let's dive into simplifying this radical expression. We have:

$$\frac{5 \sqrt{18}-2 \sqrt{50}}{\sqrt{50}}$$

This might look a bit intimidating at first, but don't worry, we'll break it down step by step. The key here is to simplify the radicals individually and then see what we can do. Radicals, especially square roots, are a fundamental part of mathematics, appearing in various areas from geometry to algebra. Simplifying them not only makes calculations easier but also helps in understanding the underlying mathematical concepts. When tackling expressions involving radicals, the initial focus should always be on breaking down the numbers inside the square roots into their prime factors. This allows us to identify perfect squares, which can then be extracted from the radical, simplifying the expression. Additionally, rationalizing the denominator is another crucial technique. It involves eliminating any radicals from the denominator, often by multiplying both the numerator and denominator by a suitable expression. This step is essential for standardizing the form of the expression and making it easier to compare or combine with other expressions. Furthermore, understanding the properties of radicals, such as the product and quotient rules, can significantly aid in simplifying complex expressions. These rules allow us to combine or separate radicals, making the manipulation of expressions more straightforward. Remember, practice is key to mastering radical simplification. The more you work with these expressions, the more comfortable and confident you'll become in handling them. Always look for opportunities to apply these techniques, and don't hesitate to break down problems into smaller, manageable steps. By following this approach, you'll find that simplifying radicals becomes a natural and intuitive process.

Step 1: Simplify the Radicals

First, let's simplify $\sqrt{18}$ and $\sqrt{50}$. We need to find the largest perfect square factors of 18 and 50. For $\sqrt{18}$, we can write 18 as $9 \times 2$, and since 9 is a perfect square ($3^2$), we have:

$$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$$

Now let's simplify $\sqrt{50}$. We can write 50 as $25 \times 2$, and since 25 is a perfect square ($5^2$), we get:

$$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$$

Breaking down radicals into their simplest forms is a crucial skill in mathematics. It allows us to manipulate and combine terms more easily, ultimately leading to simplified expressions. When simplifying radicals, we are essentially trying to find the largest perfect square factor within the radicand (the number inside the square root). This process involves identifying factors that are perfect squares (e.g., 4, 9, 16, 25, etc.) and then extracting their square roots. For instance, when simplifying $\sqrt{18}$, we recognize that 18 can be factored into $9 \times 2$, where 9 is a perfect square. Taking the square root of 9 gives us 3, leaving us with $3\sqrt{2}$. Similarly, for $\sqrt{50}$, we factor it into $25 \times 2$, where 25 is a perfect square. The square root of 25 is 5, resulting in $5\sqrt{2}$. This method not only simplifies the radicals but also makes it easier to combine like terms if there are multiple radicals in an expression. Remember, the goal is to reduce the radicand to the smallest possible number while expressing the radical in its simplest form. Practice makes perfect when it comes to simplifying radicals. The more you work with different numbers and expressions, the quicker you'll become at identifying perfect square factors and simplifying radicals efficiently. This skill is essential for more advanced topics in algebra and calculus, so mastering it early on will greatly benefit your mathematical journey.

Step 2: Substitute the Simplified Radicals

Now, let's substitute these simplified radicals back into the original expression:

$$\frac{5 \sqrt{18}-2 \sqrt{50}}{\sqrt{50}} = \frac{5(3\sqrt{2})-2(5\sqrt{2})}{5\sqrt{2}}$$

This substitution step is a crucial bridge between simplifying individual radical terms and simplifying the entire expression. By replacing the original radicals with their simplified forms, we make the expression easier to manipulate and ultimately solve. The process involves taking the simplified radicals, which we found in the previous step, and plugging them back into the original equation. This ensures that we are working with the most basic forms of the radicals, making subsequent operations more straightforward. For example, in our case, we replaced $\sqrt{18}$ with $3\sqrt{2}$ and $\sqrt{50}$ with $5\sqrt{2}$. This substitution transforms the expression into one where the radical terms are more manageable. It allows us to combine like terms and perform other algebraic operations more effectively. This step is not just about substituting values; it's about transforming the expression into a form that is easier to work with. It sets the stage for the next steps in the simplification process, where we can apply algebraic principles to further reduce the expression. Moreover, this substitution highlights the importance of simplifying radicals early in the process. By simplifying each radical term first, we reduce the complexity of the overall expression, making the substitution process smoother and the subsequent steps more efficient. In essence, this step is a pivotal part of the simplification strategy, ensuring that we are dealing with the most basic forms of the radicals before moving forward.

Step 3: Simplify the Numerator

Let's simplify the numerator:

$$5(3\sqrt{2})-2(5\sqrt{2}) = 15\sqrt{2} - 10\sqrt{2}$$

Now, we can combine these terms since they both have $\sqrt{2}$:

$$15\sqrt{2} - 10\sqrt{2} = (15-10)\sqrt{2} = 5\sqrt{2}$$

Simplifying the numerator is a critical step in reducing complex fractions and expressions. It involves applying the basic rules of arithmetic and algebra to combine like terms and reduce the expression to its simplest form. This step is often the key to unlocking further simplification in the overall problem. In the context of expressions with radicals, simplifying the numerator typically means combining terms that have the same radical part. For example, in our case, we had $15\sqrt{2} - 10\sqrt{2}$. Since both terms contain $\sqrt{2}$, they are considered like terms and can be combined. This combination is achieved by simply subtracting the coefficients (the numbers in front of the radical), which in this case are 15 and 10. The result is $5\sqrt{2}$. This process is similar to combining like terms in algebraic expressions, such as $5x - 3x$, where we would subtract the coefficients to get $2x$. The underlying principle is the same: we are grouping and combining terms that share a common factor, whether it's a variable or a radical. Simplifying the numerator not only makes the expression easier to read and understand but also sets the stage for further simplification. In many cases, simplifying the numerator can reveal common factors with the denominator, allowing us to reduce the fraction to its simplest form. Therefore, mastering the technique of simplifying the numerator is essential for anyone looking to excel in algebra and beyond. It's a fundamental skill that lays the groundwork for more advanced mathematical concepts and problem-solving strategies.

Step 4: Final Simplification

Now, we have:

$$\frac{5\sqrt{2}}{5\sqrt{2}}$$

Anything divided by itself is 1 (as long as it's not zero), so:

$$\frac{5\sqrt{2}}{5\sqrt{2}} = 1$$

Final simplification is the ultimate goal in any mathematical problem, representing the culmination of all previous steps. It's the process of reducing an expression to its most basic and concise form, where no further operations can be performed. This step not only provides the answer but also demonstrates a clear understanding of the mathematical principles involved. In the context of fractions, final simplification often involves canceling out common factors in the numerator and the denominator. This can be achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, if we have a fraction like $\frac{10}{15}$, we can simplify it by dividing both numbers by their GCD, which is 5. This gives us $\frac{2}{3}$, the simplest form of the fraction. Similarly, when dealing with expressions involving radicals, the final simplification might involve canceling out common radical terms. In our example, we had $\frac{5\sqrt{2}}{5\sqrt{2}}$. Here, the entire term $5\sqrt{2}$ is present in both the numerator and the denominator. Since any non-zero number divided by itself is 1, we can cancel out the entire term, leaving us with the simplified answer of 1. This final step is not just about getting the correct answer; it's also about showcasing the elegance and efficiency of mathematical operations. It demonstrates the ability to apply the rules and principles of mathematics to reduce a complex expression to its simplest, most understandable form. Mastering the art of final simplification is crucial for success in mathematics, as it ensures that solutions are not only correct but also presented in the most concise and clear manner possible.

Answer

So, the simplified expression is 1.

Therefore, the correct answer is A. 1.

Simplifying radical expressions might seem tricky at first, but with practice, you'll get the hang of it. Remember to break down the radicals, substitute, simplify the numerator and denominator, and then do the final simplification. Keep practicing, and you'll ace these problems in no time! You got this! This problem highlights the importance of understanding the properties of radicals and how to manipulate them. Remember to always look for the largest perfect square factor when simplifying radicals, and don't be afraid to break the problem down into smaller steps. Good luck with your studies, and keep simplifying!