Simplify (8+√288)/4: A Step-by-Step Guide

Hey guys! Today, we're going to break down how to simplify the radical expression $\frac{8+\sqrt{288}}{4}$. Don't worry, it might look a little intimidating at first, but we'll take it step by step and you'll see it's totally manageable. Our goal is to write the answer in its exact form, meaning no decimals allowed! We'll use simplified radicals instead. So, let's dive in!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what we're dealing with. The expression we need to simplify is $\frac{8+\sqrt{288}}{4}$. This involves a square root, addition, and division. Remember, the key to simplifying radical expressions is to find perfect square factors within the radical. This allows us to pull out those factors and make the expression cleaner and easier to work with. Radical expressions often seem complex, but they follow clear mathematical rules. Our initial expression, $\frac{8+\sqrt{288}}{4}$, combines a whole number (8) with a radical term ($\sqrt{288}$) in the numerator, all divided by 4. To simplify this, we aim to reduce the radical term and then see if we can further simplify the entire fraction. Simplifying radicals involves identifying perfect square factors within the radicand (the number inside the square root). A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). By finding these factors, we can rewrite the radical in a simpler form. For instance, $\sqrt{16}$ simplifies to 4 because 16 is a perfect square. In our case, we need to find the largest perfect square factor of 288. Once the radical is simplified, we’ll look for common factors between the terms in the numerator and the denominator. This is similar to reducing regular fractions. If there are common factors, we divide each term by the greatest common factor to get the simplest form of the expression. This process ensures that the final answer is in its most reduced and easiest-to-understand format. Throughout the simplification process, we'll avoid decimals and keep the answer in its exact form, using simplified radicals. This means we'll work with whole numbers and radicals, rather than converting radicals to decimal approximations. This approach is crucial in many mathematical contexts, as it preserves the precise value of the expression.

Step 1: Simplify the Square Root

The heart of simplifying this expression lies in simplifying the square root, $\sqrt{288}$. To do this, we need to find the largest perfect square that divides 288. Think of perfect squares like 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. Which of these divides 288? It might take a little trial and error, but you'll find that 144 is the largest perfect square factor of 288 because $288 = 144 \times 2$. So, we can rewrite $\sqrt{288}$ as $\sqrt{144 \times 2}$. Now, remember the property of square roots that says $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. Applying this, we get $\sqrt{144 \times 2} = \sqrt{144} \times \sqrt{2}$. We know that $\sqrt{144} = 12$, so our expression simplifies to $12\sqrt{2}$. This is a major step forward! Simplifying the square root of 288 is crucial for solving the entire expression. The initial challenge is to identify the largest perfect square factor within 288. Perfect squares are numbers that result from squaring an integer (e.g., $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, and so on). Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc. By systematically checking these numbers, we find that 144 is the largest perfect square that divides 288 evenly. Specifically, $288 = 144 \times 2$. Once we identify the perfect square factor, we can rewrite the square root of 288 as the square root of 144 times 2, or $\sqrt{144 \times 2}$. Next, we use the property of square roots that allows us to separate the factors under the radical: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. Applying this property, $\sqrt{144 \times 2}$ becomes $\sqrt{144} \times \sqrt{2}$. Since 144 is a perfect square, its square root is an integer. The square root of 144 is 12, so we have $\sqrt{144} = 12$. Thus, the expression $\sqrt{144} \times \sqrt{2}$ simplifies to $12\sqrt{2}$. This means that $\sqrt{288}$ can be simplified to $12\sqrt{2}$. This simplified form is much easier to work with in the overall expression.

Step 2: Substitute Back into the Original Expression

Now that we've simplified $\sqrt{288}$ to $12\sqrt{2}$, we can substitute this back into our original expression: $\frac{8+\sqrt{288}}{4}$ becomes $\frac{8+12\sqrt{2}}{4}$. See? It's already looking simpler! We've replaced the more complex radical with its simplified form. Substituting back into the original expression is a key step in solving the problem. We started with $\frac{8+\sqrt{288}}{4}$, and after simplifying $\sqrt{288}$ to $12\sqrt{2}$, we substitute this back into the original expression. This gives us $\frac{8+12\sqrt{2}}{4}$. This substitution is crucial because it replaces the complex radical term with its simplified equivalent, making the entire expression easier to manage. The new expression, $\frac{8+12\sqrt{2}}{4}$, consists of a numerator with two terms: a whole number (8) and a simplified radical term ($12\sqrt{2}$). Both of these terms are now easily divisible by a common factor, which is the next step in simplifying the expression. This substitution sets the stage for further simplification by allowing us to focus on reducing the fraction. By replacing the original radical with its simplified form, we have made the expression more approachable and easier to work with. This step is a perfect example of how breaking down a problem into smaller, manageable parts can lead to a solution. The substituted expression retains the exact value of the original but is now in a form that allows us to proceed with simplification. This highlights the importance of simplification in mathematics – it makes complex problems more accessible and solvable.

Step 3: Simplify the Fraction

We now have $\frac{8+12\sqrt{2}}{4}$. Notice that both 8 and 12 are divisible by 4. This means we can simplify the fraction by dividing each term in the numerator by the denominator, 4. So, $\frac{8}{4} = 2$ and $\frac{12\sqrt{2}}{4} = 3\sqrt{2}$. Therefore, our expression simplifies to $2 + 3\sqrt{2}$. And that's it! We've simplified the radical expression. The final simplification involves dividing each term in the numerator by the denominator. In our expression, $\frac{8+12\sqrt{2}}{4}$, both the whole number term (8) and the radical term ($12\sqrt{2}$) can be divided by 4. This is possible because 4 is a common factor of both 8 and 12. Dividing 8 by 4 gives us 2. This is a straightforward arithmetic operation. Next, we divide $12\sqrt{2}$ by 4. To do this, we divide the coefficient (12) by 4, leaving the $\sqrt{2}$ part unchanged. So, $\frac{12\sqrt{2}}{4} = 3\sqrt{2}$. Now, combining these simplified terms, we have $2 + 3\sqrt{2}$. This is the simplified form of the original expression. There are no more common factors to reduce, and the radical is in its simplest form. The expression $2 + 3\sqrt{2}$ is the exact form of the simplified expression, meaning it does not involve any decimal approximations. This is important in many mathematical contexts where precise values are necessary. Simplifying the fraction in this way ensures that the final answer is in its most reduced and easiest-to-understand format. The process of dividing each term in the numerator by the denominator is a common technique in simplifying algebraic fractions. It allows us to reduce the fraction to its simplest form, making it easier to work with in further calculations or applications.

Final Answer

So, the simplified form of $\frac{8+\sqrt{288}}{4}$ is $2 + 3\sqrt{2}$. You can enter this as 2+3sqrt(2). Great job, guys! You've successfully simplified a radical expression. Remember to always look for perfect square factors and simplify fractions whenever possible. Keep practicing, and you'll become a pro at simplifying radicals! The final answer, $2 + 3\sqrt{2}$, represents the simplified form of the original expression $\frac{8+\sqrt{288}}{4}$. This answer is in its exact form, meaning it does not include any decimal approximations. The expression consists of a whole number (2) and a radical term ($3\sqrt{2}$). The radical term is simplified, as the square root of 2 cannot be further reduced. The process of simplifying the radical expression involved several key steps. First, we identified the largest perfect square factor of 288, which was 144. This allowed us to rewrite $\sqrt{288}$ as $\sqrt{144 \times 2}$, which simplified to $12\sqrt{2}$. Next, we substituted this simplified radical back into the original expression, resulting in $\frac{8+12\sqrt{2}}{4}$. Finally, we simplified the fraction by dividing each term in the numerator by the denominator, 4. This gave us the simplified expression $2 + 3\sqrt{2}$. This final answer is the most reduced form of the original expression, and it is essential in many mathematical contexts where precise values are needed. To express this answer using the notation mentioned in the original problem, you would enter it as 2+3sqrt(2). This notation is commonly used in mathematical software and online platforms to represent radical expressions. The process of simplifying radical expressions is a fundamental skill in algebra and is crucial for solving more complex mathematical problems. By breaking down the problem into smaller steps, we can systematically simplify the expression and arrive at the final answer.