Simplifying Radical Expressions A Step-by-Step Guide

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. Radicals, often referred to as square roots, cube roots, and so on, can appear daunting at first glance. However, by understanding the underlying principles of radical simplification, we can transform complex expressions into their simplest forms. This article delves into the process of simplifying the expression $5 \sqrt{28}-7 \sqrt{12}+2 \sqrt{75}-2 \sqrt{7}$, providing a step-by-step guide to mastering this essential mathematical technique.

Understanding Radicals

Before we dive into the simplification process, let's first establish a solid understanding of radicals. A radical is a mathematical expression that involves a root, such as a square root (√), cube root (∛), or higher-order roots. The number under the radical symbol is called the radicand, and the small number above the radical symbol (if present) is called the index. For instance, in the expression √9, the radical symbol is √, the radicand is 9, and the index is 2 (since it's a square root). Similarly, in the expression ∛8, the radical symbol is ∛, the radicand is 8, and the index is 3 (since it's a cube root).

The square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3, because 3 * 3 = 9. The cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 = 8. Higher-order roots follow the same principle.

Simplifying radicals involves expressing them in their simplest form, where the radicand has no perfect square factors (for square roots), perfect cube factors (for cube roots), and so on. This process often involves factoring the radicand and extracting any perfect square, cube, or higher-order factors. Mastering the simplification of radicals is crucial for various mathematical operations, including solving equations, simplifying expressions, and performing calculus.

Step-by-Step Simplification of $5 \sqrt{28}-7 \sqrt{12}+2 \sqrt{75}-2 \sqrt{7}$

Now, let's tackle the expression $5 \sqrt{28}-7 \sqrt{12}+2 \sqrt{75}-2 \sqrt{7}$ step-by-step. Our goal is to simplify each radical term individually and then combine like terms.

Step 1: Simplify $\sqrt{28}$

To simplify $\sqrt{28}$, we need to find the prime factorization of 28. The prime factorization of 28 is 2 * 2 * 7, which can be written as $2^2 * 7$. Therefore, we can rewrite $\sqrt{28}$ as $\sqrt{2^2 * 7}$.

Using the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we can separate the radical: $\sqrt{2^2 * 7} = \sqrt{2^2} * \sqrt{7}$.

Since $\sqrt{2^2} = 2$, the simplified form of $\sqrt{28}$ is $2\sqrt{7}$. So, the first term becomes $5 \sqrt{28} = 5 * 2\sqrt{7} = 10\sqrt{7}$. This first step is crucial for the simplification process, as we're breaking down the radical into its simplest components, preparing it for further manipulation and combination with other terms.

Step 2: Simplify $\sqrt{12}$

Next, we simplify $\sqrt{12}$. The prime factorization of 12 is 2 * 2 * 3, which can be written as $2^2 * 3$. Thus, we can rewrite $\sqrt{12}$ as $\sqrt{2^2 * 3}$.

Applying the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we separate the radical: $\sqrt{2^2 * 3} = \sqrt{2^2} * \sqrt{3}$.

Since $\sqrt{2^2} = 2$, the simplified form of $\sqrt{12}$ is $2\sqrt{3}$. So, the second term becomes $-7 \sqrt{12} = -7 * 2\sqrt{3} = -14\sqrt{3}$. This step, similar to the first, involves breaking down the radical into its constituent factors, extracting any perfect squares to simplify the expression and make it easier to work with in subsequent steps.

Step 3: Simplify $\sqrt{75}$

Now, let's simplify $\sqrt{75}$. The prime factorization of 75 is 3 * 5 * 5, which can be written as $3 * 5^2$. Therefore, we can rewrite $\sqrt{75}$ as $\sqrt{3 * 5^2}$.

Using the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we separate the radical: $\sqrt{3 * 5^2} = \sqrt{3} * \sqrt{5^2}$.

Since $\sqrt{5^2} = 5$, the simplified form of $\sqrt{75}$ is $5\sqrt{3}$. So, the third term becomes $2 \sqrt{75} = 2 * 5\sqrt{3} = 10\sqrt{3}$. Again, we're employing the same strategy of prime factorization and separation of radicals to simplify the term, allowing us to express it in a more manageable form for the final simplification process.

Step 4: Simplify $\sqrt{7}$

The radical $\sqrt{7}$ is already in its simplest form because 7 is a prime number and has no perfect square factors other than 1. Therefore, the fourth term, $-2 \sqrt{7}$, remains as it is. This step highlights an important aspect of radical simplification: recognizing when a radical is already in its simplest form, which often occurs when the radicand is a prime number or has no perfect square factors.

Step 5: Combine Like Terms

Now that we have simplified each radical term, we can substitute the simplified forms back into the original expression:

$5 \sqrt{28}-7 \sqrt{12}+2 \sqrt{75}-2 \sqrt{7} = 10\sqrt{7} - 14\sqrt{3} + 10\sqrt{3} - 2\sqrt{7}$.

To combine like terms, we group the terms with the same radical: $(10\sqrt{7} - 2\sqrt{7}) + (-14\sqrt{3} + 10\sqrt{3})$.

Combining the coefficients of the like terms, we get: $8\sqrt{7} - 4\sqrt{3}$.

Therefore, the simplified form of the expression $5 \sqrt{28}-7 \sqrt{12}+2 \sqrt{75}-2 \sqrt{7}$ is $8\sqrt{7} - 4\sqrt{3}$. This final step brings together all the individual simplifications, combining like terms to arrive at the most concise and simplified form of the original expression.

Conclusion

Simplifying radical expressions is a fundamental skill in mathematics. By understanding the properties of radicals, prime factorization, and the process of combining like terms, we can effectively simplify complex expressions into their simplest forms. In this article, we successfully simplified the expression $5 \sqrt{28}-7 \sqrt{12}+2 \sqrt{75}-2 \sqrt{7}$ to $8\sqrt{7} - 4\sqrt{3}$, demonstrating the step-by-step process involved in radical simplification. Mastering these techniques is essential for success in various mathematical fields, including algebra, calculus, and beyond. The ability to manipulate and simplify radicals not only enhances problem-solving skills but also provides a deeper understanding of mathematical concepts.