Simplifying Radical Expressions $2 \sqrt{48} + \sqrt{200} - \sqrt{75} - 4 \sqrt{32}$

Understanding Radical Expressions

Before we dive into simplifying the given expression, let's briefly discuss radical expressions. A radical expression is a mathematical expression containing a radical symbol (√), which indicates the root of a number. The most common radical is the square root, denoted by √, which asks, "What number, when multiplied by itself, equals the number under the radical?" Simplifying radical expressions involves reducing them to their simplest form, where the number under the radical (the radicand) has no perfect square factors other than 1.

Simplifying radical expressions is a fundamental skill in algebra, and it's crucial for various mathematical operations, including solving equations, working with geometric figures, and more. When we talk about simplifying these expressions, we essentially aim to rewrite them in a form where the radicand (the number under the square root) has no perfect square factors other than 1. This process often involves breaking down the radicand into its prime factors and identifying pairs of identical factors. Each pair can then be taken out of the square root as a single factor. This not only makes the expression cleaner but also makes it easier to compare and combine like terms, which is essential for further calculations.

The practical applications of simplifying radical expressions are vast. In geometry, for instance, when calculating the lengths of sides in right-angled triangles using the Pythagorean theorem, we often end up with radical expressions. Simplifying these expressions gives us a more precise and manageable form of the answer. In physics, many formulas involve square roots, such as those dealing with energy and motion. Simplifying these roots allows for easier calculations and a clearer understanding of the relationships between different physical quantities. Moreover, in advanced mathematics, such as calculus and complex analysis, simplifying radicals is a routine step in various problem-solving techniques. Therefore, mastering this skill is not just about handling algebraic expressions; it’s about building a solid foundation for more advanced mathematical and scientific concepts.

Breaking Down the Problem: $2 \sqrt{48} + \sqrt{200} - \sqrt{75} - 4 \sqrt{32}$

Let's tackle the given expression step by step: $2 \sqrt{48} + \sqrt{200} - \sqrt{75} - 4 \sqrt{32}$. Our goal is to simplify each radical term individually and then combine any like terms.

The first term we encounter is $2 \sqrt{48}$. To simplify this, we need to find the largest perfect square that divides 48. The perfect squares less than 48 are 1, 4, 9, 16, 25, and 36. Among these, 16 is the largest perfect square that divides 48. So, we can rewrite $\sqrt{48}$ as $\sqrt{16 \times 3}$. Applying the property of square roots that states $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we get $\sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$. Now, multiplying this by the coefficient 2, we have $2 \times 4\sqrt{3} = 8\sqrt{3}$.

Next, we simplify $\sqrt{200}$. The perfect squares less than 200 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and 196. The largest perfect square that divides 200 is 100. Thus, we rewrite $\sqrt{200}$ as $\sqrt{100 \times 2}$. Applying the same property as before, we get $\sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}$. This term is now simplified.

For the term $-\sqrt{75}$, we identify 25 as the largest perfect square that divides 75. So, we rewrite $\sqrt{75}$ as $\sqrt{25 \times 3}$. This simplifies to $\sqrt{25} \times \sqrt{3} = 5\sqrt{3}$. Therefore, $-\sqrt{75}$ becomes $-5\sqrt{3}$.

Finally, we tackle $-4\sqrt{32}$. The largest perfect square that divides 32 is 16. We rewrite $\sqrt{32}$ as $\sqrt{16 \times 2}$, which simplifies to $\sqrt{16} \times \sqrt{2} = 4\sqrt{2}$. Multiplying by the coefficient -4, we get $-4 \times 4\sqrt{2} = -16\sqrt{2}$.

Step-by-Step Simplification

Let's break down the simplification process of the expression $2 \sqrt{48} + \sqrt{200} - \sqrt{75} - 4 \sqrt{32}$ into manageable steps:

1. Simplify $2\sqrt{48}$:
   • Identify the largest perfect square factor of 48, which is 16.
   • Rewrite $\sqrt{48}$ as $\sqrt{16 \times 3}$.
   • Apply the square root property: $\sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$.
   • Multiply by the coefficient: $2 \times 4\sqrt{3} = 8\sqrt{3}$.
2. Simplify $\sqrt{200}$:
   • Identify the largest perfect square factor of 200, which is 100.
   • Rewrite $\sqrt{200}$ as $\sqrt{100 \times 2}$.
   • Apply the square root property: $\sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}$.
3. Simplify $-\sqrt{75}$:
   • Identify the largest perfect square factor of 75, which is 25.
   • Rewrite $\sqrt{75}$ as $\sqrt{25 \times 3}$.
   • Apply the square root property: $\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$.
   • Include the negative sign: $-\sqrt{75} = -5\sqrt{3}$.
4. Simplify $-4\sqrt{32}$:
   • Identify the largest perfect square factor of 32, which is 16.
   • Rewrite $\sqrt{32}$ as $\sqrt{16 \times 2}$.
   • Apply the square root property: $\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$.
   • Multiply by the coefficient: $-4 \times 4\sqrt{2} = -16\sqrt{2}$.

Combining Like Terms

After simplifying each radical term individually, we have:

• $2\sqrt{48} = 8\sqrt{3}$
• $\sqrt{200} = 10\sqrt{2}$
• $-\sqrt{75} = -5\sqrt{3}$
• $-4\sqrt{32} = -16\sqrt{2}$

Now, we can rewrite the original expression as:

$8\sqrt{3} + 10\sqrt{2} - 5\sqrt{3} - 16\sqrt{2}$

To simplify further, we combine like terms. Like terms are those that have the same radicand (the number inside the square root). In this case, we have terms with $\sqrt{3}$ and terms with $\sqrt{2}$.

Combining the terms with $\sqrt{3}$, we have:

$8\sqrt{3} - 5\sqrt{3} = (8 - 5)\sqrt{3} = 3\sqrt{3}$

Combining the terms with $\sqrt{2}$, we have:

$10\sqrt{2} - 16\sqrt{2} = (10 - 16)\sqrt{2} = -6\sqrt{2}$

Final Simplification

After combining like terms, we have:

$3\sqrt{3} - 6\sqrt{2}$

This expression is now in its simplest form because there are no more like terms to combine, and the radicands (3 and 2) have no perfect square factors other than 1. Therefore, the simplified form of the given radical expression is $3\sqrt{3} - 6\sqrt{2}$. This final simplified form is not only concise but also makes it easier to use in further calculations or comparisons.

In summary, simplifying radical expressions involves breaking down the radicands into their prime factors, identifying and extracting perfect square factors, and then combining like terms. This process not only simplifies the expression but also enhances our understanding of the underlying mathematical structure. Mastering this skill is crucial for success in algebra and beyond, as it appears in various contexts across mathematics and science. By following a systematic approach, we can confidently simplify even the most complex radical expressions, paving the way for more advanced mathematical explorations.

Therefore, the final answer is:

Answer:

$3\sqrt{3} - 6\sqrt{2}$