Simplify: $2 \sqrt{9} - 4 + \sqrt{3}$ | Math Problem

In this article, we'll walk through the process of simplifying the mathematical expression $2 \sqrt{9} - 4 + \sqrt{3}$. This involves understanding basic arithmetic operations and square roots. Let's break it down step by step to make it easy to follow. This problem combines both simplification of square roots and basic arithmetic. A strong foundation in these areas is crucial for tackling more complex mathematical problems. By the end of this guide, you’ll not only be able to solve this specific problem but also gain confidence in handling similar expressions. Understanding the properties of square roots, such as how to simplify perfect squares, is essential. Additionally, knowing the order of operations (PEMDAS/BODMAS) ensures you perform calculations in the correct sequence. Let’s dive in and simplify this expression together!

Understanding the Expression

The expression we need to simplify is $2 \sqrt{9} - 4 + \sqrt{3}$. This expression contains three terms: $2 \sqrt{9}$, $-4$, and $\sqrt{3}$. The first term involves a square root, the second is a constant, and the third term also involves a square root. Our goal is to simplify each term as much as possible and then combine like terms. Before we start simplifying, it's essential to understand each component of the expression. The term $2 \sqrt{9}$ means 2 times the square root of 9. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we need to find a number that, when multiplied by itself, equals 9. The term $-4$ is a constant, which means it's a fixed value and doesn't change. The term $\sqrt{3}$ represents the square root of 3. Since 3 is not a perfect square, we will leave it as a square root in our simplified expression. Understanding these components is crucial for simplifying the entire expression correctly. Now, let’s move on to the next step where we simplify the square root of 9. This will help us further simplify the expression and move closer to our final answer. Remember, breaking down the expression into smaller, manageable parts makes it easier to solve.

Step 1: Simplify the Square Root

The first step in simplifying the expression $2 \sqrt{9} - 4 + \sqrt{3}$ is to simplify the square root, specifically $\sqrt{9}$. We need to find a number that, when multiplied by itself, equals 9. The square root of 9 is 3 because $3 \times 3 = 9$. Now we can replace $\sqrt{9}$ with 3 in our expression. So, our expression becomes $2 \times 3 - 4 + \sqrt{3}$. Simplifying square roots is a fundamental skill in algebra, and recognizing perfect squares like 9 can significantly speed up the simplification process. Perfect squares are numbers that have integers as their square roots, such as 1, 4, 9, 16, 25, and so on. Being familiar with these numbers can help you quickly simplify expressions involving square roots. Additionally, understanding that the square root of a number is the value that, when squared, gives you the original number is essential. In this case, since $3^2 = 9$, $\sqrt{9} = 3$. Now that we've simplified the square root, we can move on to the next step, which involves performing the multiplication operation. This will further simplify our expression and bring us closer to our final answer. Remember, each step in simplifying an expression builds on the previous one, so it's important to understand each step thoroughly before moving on.

Step 2: Perform the Multiplication

After simplifying the square root, our expression is now $2 \times 3 - 4 + \sqrt{3}$. The next step is to perform the multiplication operation. We need to multiply 2 by 3, which gives us 6. So, our expression becomes $6 - 4 + \sqrt{3}$. Multiplication is a basic arithmetic operation, but it's crucial to perform it correctly to ensure the accurate simplification of the expression. In this case, we are simply multiplying two integers, 2 and 3. This step is relatively straightforward, but it's essential to follow the correct order of operations, which dictates that multiplication should be performed before addition and subtraction. Now that we've completed the multiplication, we can move on to the next step, which involves combining the constants in the expression. This will further simplify our expression and bring us closer to our final answer. Remember, each step in simplifying an expression is important, and performing them in the correct order is crucial for achieving the correct result. Now, let's proceed to combine the constants and see how the expression simplifies further.

Step 3: Combine the Constants

Now that we have $6 - 4 + \sqrt{3}$, we need to combine the constants. The constants in this expression are 6 and -4. Subtracting 4 from 6 gives us 2. So, the expression becomes $2 + \sqrt{3}$. Combining constants is a fundamental step in simplifying algebraic expressions. It involves adding or subtracting the numerical values that do not have any variables attached to them. In this case, we are simply subtracting 4 from 6, which is a basic arithmetic operation. However, it's important to pay attention to the signs of the numbers. Here, we have a positive 6 and a negative 4, so we subtract 4 from 6 to get 2. Now that we've combined the constants, we have simplified the expression to $2 + \sqrt{3}$. This is the simplest form of the expression because we cannot further simplify the square root of 3 or combine it with the constant 2. Therefore, we have reached our final answer. Remember, combining constants is an essential skill in algebra, and mastering it will help you simplify more complex expressions. Now that we've completed all the steps, let's review the entire process to ensure we understand each step thoroughly.

Final Answer

After simplifying the expression $2 \sqrt{9} - 4 + \sqrt{3}$, we arrive at the final answer: $2 + \sqrt{3}$. We started by simplifying the square root of 9, then performed the multiplication, and finally combined the constants. Each step was crucial in simplifying the expression to its simplest form. Therefore, $2 \sqrt{9} - 4 + \sqrt{3} = 2 + \sqrt{3}$. Understanding each step in the simplification process is essential for mastering algebraic expressions. We began by identifying the components of the expression, then simplified each component individually, and finally combined like terms. This approach can be applied to a wide range of algebraic expressions, making it a valuable skill to develop. By following these steps, you can confidently simplify similar expressions and improve your understanding of algebra. Remember to always pay attention to the order of operations and the properties of square roots to ensure accurate simplification. Now that we've reached the final answer, let's take a moment to review the entire process and reinforce our understanding.

In summary, we simplified the expression $2 \sqrt{9} - 4 + \sqrt{3}$ by first recognizing that $\sqrt{9} = 3$. We then multiplied 2 by 3 to get 6, resulting in the expression $6 - 4 + \sqrt{3}$. Next, we combined the constants 6 and -4 to get 2, leading to the final simplified form of $2 + \sqrt{3}$. This step-by-step approach allowed us to break down the problem into manageable parts and arrive at the solution efficiently. Understanding these fundamental steps is essential for tackling more complex algebraic problems. Make sure to practice similar problems to reinforce your understanding and improve your problem-solving skills. Remember, consistent practice is key to mastering algebra and building confidence in your mathematical abilities.

For more information on simplifying expressions, check out Khan Academy's resource on the order of operations.