Solving Radical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the world of radical expressions! Don't worry, it might seem a bit intimidating at first, but with a little practice and the right approach, you'll be solving these problems like a pro. This guide will walk you through calculating several complex radical expressions step-by-step. We'll break down each problem, explaining the logic behind every move, so you can understand the process and apply it to other similar challenges. Get ready to flex those math muscles!

Understanding the Basics: Simplifying Radicals

Before we jump into the calculations, let's refresh some fundamental concepts about simplifying radicals. A radical expression is simply an expression that contains a radical symbol (√), also known as a square root. Simplifying a radical involves reducing the expression under the radical sign to its simplest form. This often involves extracting perfect squares from the radical. For example, the square root of 9 (√9) is 3, because 3 * 3 = 9. Likewise, √16 = 4, √25 = 5, and so on.

To simplify radicals, we look for perfect square factors within the radicand (the number under the radical). When we find a perfect square factor, we can take its square root and move it outside the radical sign. This process makes the expression cleaner and easier to work with. For example, let's simplify √20. We know that 20 = 4 * 5, and 4 is a perfect square. Thus, √20 = √(4 * 5) = √4 * √5 = 2√5. This is the simplified form. Another important aspect to keep in mind is the properties of radicals: √a * √b = √(a * b). This rule allows us to combine or separate radical expressions, which is key to simplifying and solving problems involving radicals. Remembering and applying these rules are absolutely essential to solving the problems that we are about to tackle. Are you ready?

So, let's put these concepts into practice. Get your pencils and calculators ready, and let's get solving!

Problem A: Detailed Solution

Let's tackle the first problem: a $4 \sqrt{3} \cdot(2 \sqrt{75}-\sqrt{27}+3 \sqrt{12})-2 \sqrt{3} \cdot \sqrt{48}$. This problem looks complex at first glance, but we'll break it down step-by-step, simplifying each part of the expression. Remember, always start with the most complex part of the problem. We start by simplifying each of the radicals inside the parentheses: √75, √27, and √12.

• Simplifying √75: We can rewrite 75 as 25 * 3, and since 25 is a perfect square, √75 = √(25 * 3) = √25 * √3 = 5√3.
• Simplifying √27: We can rewrite 27 as 9 * 3, where 9 is a perfect square. So, √27 = √(9 * 3) = √9 * √3 = 3√3.
• Simplifying √12: We can rewrite 12 as 4 * 3, where 4 is a perfect square. So, √12 = √(4 * 3) = √4 * √3 = 2√3.
• Simplifying √48: Similarly, we can rewrite 48 as 16 * 3, where 16 is a perfect square. So, √48 = √(16 * 3) = √16 * √3 = 4√3.

Now, let's substitute these simplified radicals back into the original expression: $4 \sqrt{3} \cdot(2(5 \sqrt{3})-3 \sqrt{3}+3(2 \sqrt{3}))-2 \sqrt{3} \cdot 4 \sqrt{3}$.

Next, simplify the terms inside the parentheses: $2(5√3) = 10√3$ $3(2√3) = 6√3$ So, the expression inside the parentheses becomes: $(10 \sqrt{3}-3 \sqrt{3}+6 \sqrt{3})$.

Combine like terms inside the parentheses: $10√3 - 3√3 + 6√3 = (10 - 3 + 6)√3 = 13√3$. Now, the original expression is: $4 \sqrt{3} \cdot 13 \sqrt{3} - 2 \sqrt{3} \cdot 4 \sqrt{3}$.

Now, we multiply the terms. For the first part of the expression: $4 \sqrt{3} \cdot 13 \sqrt{3} = 4 * 13 * (\sqrt{3} * \sqrt{3}) = 52 * 3 = 156$. For the second part of the expression: $2 \sqrt{3} \cdot 4 \sqrt{3} = 2 * 4 * (\sqrt{3} * \sqrt{3}) = 8 * 3 = 24$.

Finally, subtract the second result from the first result: $156 - 24 = 132$. Therefore, the solution to the first problem is 132.

*Key takeaway: Breaking down complex radicals into simpler forms is essential. Also, carefully performing operations and order of operations are the keys to a correct solution.

Problem B: Detailed Solution

Let's proceed to the second problem: b $(\sqrt{243}-2 \sqrt{3}) \cdot(\sqrt{147})+3 \sqrt{8} \cdot(5 \sqrt{72}+6 \sqrt{128})$. This problem involves multiple radicals and operations, so let’s approach it step-by-step. Remember to break down each radical expression into its simplest form before multiplying. Let's start with simplifying the radicals:

• Simplifying √243: We can rewrite 243 as 81 * 3, and 81 is a perfect square. Thus, √243 = √(81 * 3) = √81 * √3 = 9√3.
• Simplifying √147: We can rewrite 147 as 49 * 3, where 49 is a perfect square. So, √147 = √(49 * 3) = √49 * √3 = 7√3.
• Simplifying √8: We can rewrite 8 as 4 * 2, where 4 is a perfect square. So, √8 = √(4 * 2) = √4 * √2 = 2√2.
• Simplifying √72: We can rewrite 72 as 36 * 2, where 36 is a perfect square. So, √72 = √(36 * 2) = √36 * √2 = 6√2.
• Simplifying √128: We can rewrite 128 as 64 * 2, where 64 is a perfect square. So, √128 = √(64 * 2) = √64 * √2 = 8√2.

Now substitute these simplified radicals back into the expression: $(9 \sqrt{3}-2 \sqrt{3}) \cdot 7 \sqrt{3} + 3(2 \sqrt{2}) \cdot(5(6 \sqrt{2})+6(8 \sqrt{2}))$.

Simplify within the parentheses: $(9 \sqrt{3} - 2 \sqrt{3}) = 7 \sqrt{3}$. $(5(6 \sqrt{2}) + 6(8 \sqrt{2})) = (30 \sqrt{2} + 48 \sqrt{2}) = 78 \sqrt{2}$.

The expression now looks like this: $7 \sqrt{3} \cdot 7 \sqrt{3} + 6 \sqrt{2} \cdot 78 \sqrt{2}$.

Now, perform the multiplications. For the first part of the expression: $7 \sqrt{3} \cdot 7 \sqrt{3} = 7 * 7 * (\sqrt{3} * \sqrt{3}) = 49 * 3 = 147$. For the second part of the expression: $6 \sqrt{2} \cdot 78 \sqrt{2} = 6 * 78 * (\sqrt{2} * \sqrt{2}) = 468 * 2 = 936$.

Finally, add the two results: $147 + 936 = 1083$. Therefore, the solution to the second problem is 1083.

*Key takeaway: Careful simplification and organization are paramount. It is crucial to be methodical in order to avoid mistakes.

Problem C: Detailed Solution

Let's move on to the final problem: c $8 \sqrt{54}-6 \sqrt{384}+10 \sqrt{96}-5 \sqrt{216}$. This problem involves multiple terms, each with a radical, so let's break it down by simplifying each radical term individually and then combining like terms. First, let's simplify each of the radical terms:

• Simplifying √54: We can rewrite 54 as 9 * 6, and 9 is a perfect square. Thus, √54 = √(9 * 6) = √9 * √6 = 3√6. Therefore, $8 \sqrt{54} = 8 * 3 \sqrt{6} = 24 \sqrt{6}$.
• Simplifying √384: We can rewrite 384 as 64 * 6, where 64 is a perfect square. So, √384 = √(64 * 6) = √64 * √6 = 8√6. Therefore, $-6 \sqrt{384} = -6 * 8 \sqrt{6} = -48 \sqrt{6}$.
• Simplifying √96: We can rewrite 96 as 16 * 6, where 16 is a perfect square. So, √96 = √(16 * 6) = √16 * √6 = 4√6. Therefore, $10 \sqrt{96} = 10 * 4 \sqrt{6} = 40 \sqrt{6}$.
• Simplifying √216: We can rewrite 216 as 36 * 6, where 36 is a perfect square. So, √216 = √(36 * 6) = √36 * √6 = 6√6. Therefore, $-5 \sqrt{216} = -5 * 6 \sqrt{6} = -30 \sqrt{6}$.

Now, let's substitute these simplified radical terms back into the original expression: $24 \sqrt{6} - 48 \sqrt{6} + 40 \sqrt{6} - 30 \sqrt{6}$.

Combine like terms. Add or subtract the coefficients of the √6 terms: $(24 - 48 + 40 - 30) \sqrt{6}$. Calculating the coefficients: $24 - 48 = -24$; $-24 + 40 = 16$; $16 - 30 = -14$.

Thus, the result is: $-14 \sqrt{6}$. Therefore, the solution to the third problem is $-14 \sqrt{6}$.

*Key takeaway: Break down complex expressions into simpler forms by finding perfect square factors. Combine like terms for the final answer.

Conclusion: Mastering Radical Expressions

Alright, awesome work, guys! We've successfully calculated each of the radical expressions step-by-step. Remember, practice is key. The more problems you solve, the more comfortable you'll become with simplifying radicals and performing operations. Keep at it, and you'll become a master of radical expressions in no time! Keep practicing, and you'll find that these problems become easier and more enjoyable. If you have any questions or need further clarification, feel free to ask. Good luck, and keep up the great work! That's all for today!