Simplify 7√27 - 5√3 A Step-by-Step Guide

In this article, we will delve into the process of simplifying the expression $7 \sqrt{27}-5 \sqrt{3}$. This type of problem falls under the category of mathematics, specifically dealing with the simplification of radical expressions. Simplifying radical expressions is a fundamental skill in algebra and is crucial for solving more complex mathematical problems. The key to simplifying such expressions lies in identifying perfect square factors within the radicand (the number under the square root) and then extracting them. By understanding the properties of square roots and applying them systematically, we can break down complex expressions into simpler, more manageable forms. This not only makes the expression easier to understand but also facilitates further calculations and manipulations. This article will provide a step-by-step guide on how to simplify the given expression, explaining the underlying concepts and techniques involved. We'll start by understanding the properties of square roots, then move on to identifying perfect square factors, and finally, applying these concepts to simplify the given expression. Mastering this skill is essential for anyone looking to excel in algebra and related fields. Let's begin by understanding the basic properties of square roots and how they can be used to simplify expressions. The goal is to transform the expression into its simplest form, where the radicand has no more perfect square factors. This involves a combination of algebraic manipulation and a good understanding of number theory. By the end of this guide, you should be able to confidently simplify similar radical expressions and apply these techniques to other mathematical problems.

Understanding Square Roots and Simplification

To effectively simplify the expression $7 \sqrt{27}-5 \sqrt{3}$, it is crucial to first understand the fundamental properties of square roots. A square root of a number 'x' is a value that, when multiplied by itself, gives 'x'. For example, the square root of 9 is 3, because 3 * 3 = 9. The symbol for the square root is '√'. When simplifying square roots, the goal is to express the radicand (the number inside the square root symbol) in its simplest form, meaning that it should not have any perfect square factors other than 1. A perfect square is a number that can be obtained by squaring an integer (e.g., 1, 4, 9, 16, 25, etc.). The key property we'll use is the product property of square roots, which states that √(a * b) = √a * √b. This property allows us to separate the square root of a product into the product of square roots, which is crucial for simplification. For instance, if we have √12, we can rewrite it as √(4 * 3), and then apply the product property to get √4 * √3, which simplifies to 2√3. This is because 4 is a perfect square, and its square root is 2. Similarly, if we encounter an expression like √(a^2 * b), where 'a' is an integer, we can simplify it to a√b. The process of simplification involves identifying the largest perfect square factor within the radicand and then extracting its square root. This makes the expression more manageable and easier to work with. It's also important to remember that we can only combine terms with the same radicand. For example, we can combine 2√3 and 3√3 to get 5√3, but we cannot combine 2√3 and 3√2 because they have different radicands. Understanding these basic principles is essential for simplifying radical expressions effectively. Now, let's move on to applying these concepts to the given expression.

Identifying Perfect Square Factors

Before we can simplify the expression $7 \sqrt{27}-5 \sqrt{3}$, we need to identify perfect square factors within the radicands. This is a crucial step in the simplification process. In our expression, we have two terms: $7 \sqrt{27}$ and $5 \sqrt{3}$. Let's focus on the first term, $7 \sqrt{27}$. The radicand here is 27. To find perfect square factors of 27, we need to think of perfect squares that divide 27 without leaving a remainder. Perfect squares are numbers like 1, 4, 9, 16, 25, and so on. We can see that 9 is a factor of 27, as 27 = 9 * 3. And 9 is a perfect square because 9 = 3^2. So, we can rewrite $\sqrt{27}$ as $\sqrt{9 * 3}$. Now, we can apply the product property of square roots, which states that √(a * b) = √a * √b. Therefore, $\sqrt{9 * 3}$ can be rewritten as $\sqrt{9} * \sqrt{3}$. Since the square root of 9 is 3, we have 3√3. Now, let's consider the second term, $5 \sqrt{3}$. The radicand here is 3. The only perfect square factor of 3 is 1, so we cannot simplify $\sqrt{3}$ any further. It is already in its simplest form. By identifying the perfect square factor in 27, we were able to break down the first term into a simpler form. This is the key to simplifying radical expressions. We look for the largest perfect square factor to make the simplification process more efficient. In this case, 9 was the largest perfect square factor of 27. If we had chosen a smaller perfect square factor, such as 1, we wouldn't have made any progress in simplifying the expression. Now that we have identified the perfect square factors and simplified the radicals, we can move on to substituting these simplified forms back into the original expression.

Simplifying the Expression: Step-by-Step

Now that we've understood square roots and identified perfect square factors, let's simplify the expression $7 \sqrt{27}-5 \sqrt{3}$ step-by-step. First, we found that $\sqrt{27}$ can be simplified as follows:

$\sqrt{27} = \sqrt{9 * 3} = \sqrt{9} * \sqrt{3} = 3\sqrt{3}$

So, the first term of our expression, $7 \sqrt{27}$, can be rewritten as:

$7 \sqrt{27} = 7 * (3\sqrt{3}) = 21\sqrt{3}$

Now, let's look at the second term, $5 \sqrt{3}$. As we discussed earlier, $\sqrt{3}$ is already in its simplest form, so we don't need to simplify this term further. Now we can substitute the simplified form of $7 \sqrt{27}$ back into the original expression:

$7 \sqrt{27}-5 \sqrt{3} = 21\sqrt{3} - 5\sqrt{3}$

Notice that both terms now have the same radicand, $\sqrt{3}$. This means we can combine these terms by subtracting their coefficients (the numbers in front of the square root). To combine the terms, we subtract the coefficient of the second term from the coefficient of the first term:

$21\sqrt{3} - 5\sqrt{3} = (21 - 5)\sqrt{3} = 16\sqrt{3}$

Therefore, the simplified form of the expression $7 \sqrt{27}-5 \sqrt{3}$ is $16\sqrt{3}$. This step-by-step process demonstrates how to break down a complex radical expression into simpler components, identify perfect square factors, and then combine like terms. The key is to systematically apply the properties of square roots and arithmetic operations to arrive at the simplest form. By following these steps, you can confidently simplify various radical expressions.

Final Result and Conclusion

In conclusion, the simplified form of the expression $7 \sqrt{27}-5 \sqrt{3}$ is $16\sqrt{3}$. We arrived at this result by systematically breaking down the expression, identifying perfect square factors, and applying the properties of square roots. This process involved several key steps. First, we understood the fundamental properties of square roots, particularly the product property, which allows us to separate the square root of a product into the product of square roots. This property is crucial for simplifying radical expressions. Next, we identified perfect square factors within the radicands. In the term $7 \sqrt{27}$, we recognized that 27 has a perfect square factor of 9. This allowed us to rewrite $\sqrt{27}$ as $\sqrt{9 * 3}$, which then simplified to $3\sqrt{3}$. The term $5 \sqrt{3}$ was already in its simplest form, as 3 does not have any perfect square factors other than 1. Then, we substituted the simplified form of $\sqrt{27}$ back into the original expression, giving us $7 * (3\sqrt{3}) - 5\sqrt{3}$, which simplifies to $21\sqrt{3} - 5\sqrt{3}$. Finally, we combined the terms with the same radicand by subtracting their coefficients. This gave us $(21 - 5)\sqrt{3}$, which simplifies to $16\sqrt{3}$. This final result, $16\sqrt{3}$, represents the simplest form of the original expression. Simplifying radical expressions is a fundamental skill in algebra, and this example illustrates the key techniques involved. By mastering these techniques, you can confidently tackle more complex mathematical problems involving radicals. The ability to simplify radical expressions is not only useful in mathematics but also in various fields such as physics and engineering, where such expressions often arise in calculations and problem-solving.