Multiply And Simplify Radical Expressions

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In mathematics, simplifying expressions involving radicals often requires a combination of algebraic manipulation and a solid understanding of radical properties. This article delves into the process of multiplying expressions containing radicals, specifically focusing on the expression (96+27)(97βˆ’56)(9 \sqrt{6}+2 \sqrt{7})(9 \sqrt{7}-5 \sqrt{6}). We will explore the step-by-step methodology to simplify this expression, providing clear explanations and insights along the way. This comprehensive guide aims to equip you with the skills necessary to tackle similar problems with confidence and precision. The key to successfully simplifying such expressions lies in a methodical approach, applying the distributive property, and simplifying individual terms. By breaking down the problem into smaller, manageable steps, we can effectively navigate the complexities of radical expressions.

Understanding the Basics of Radicals

Before diving into the multiplication, it’s essential to grasp the fundamental properties of radicals. A radical is a root of a number, most commonly a square root (\sqrt{}) but also including cube roots, fourth roots, and so on. Simplifying radicals involves expressing them in their simplest form, where the number under the radical (the radicand) has no perfect square factors other than 1. For instance, 8\sqrt{8} can be simplified to 222\sqrt{2} because 8 has a perfect square factor of 4. Radicals can be added or subtracted only if they are like radicals, meaning they have the same radicand. For example, 323\sqrt{2} and 525\sqrt{2} are like radicals and can be combined to 828\sqrt{2}, but 323\sqrt{2} and 535\sqrt{3} cannot be directly combined. When multiplying radicals, we use the property aβ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. This property allows us to combine radicals under a single radical sign, which can then be simplified further if possible. Understanding these basic principles is crucial for effectively manipulating and simplifying expressions involving radicals. Additionally, it’s important to remember the distributive property, which states that a(b+c)=ab+aca(b + c) = ab + ac. This property is fundamental when multiplying a radical expression by another term or expression.

Step-by-Step Multiplication Process

To multiply the given expression (96+27)(97βˆ’56)(9 \sqrt{6}+2 \sqrt{7})(9 \sqrt{7}-5 \sqrt{6}), we will use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This involves multiplying each term in the first parenthesis by each term in the second parenthesis and then combining like terms. Let’s break down the process:

  1. First: Multiply the first terms in each parenthesis: (96)β‹…(97)=816β‹…7=8142(9 \sqrt{6}) \cdot (9 \sqrt{7}) = 81 \sqrt{6 \cdot 7} = 81 \sqrt{42}.
  2. Outer: Multiply the outer terms: (96)β‹…(βˆ’56)=βˆ’456β‹…6=βˆ’4536(9 \sqrt{6}) \cdot (-5 \sqrt{6}) = -45 \sqrt{6 \cdot 6} = -45 \sqrt{36}.
  3. Inner: Multiply the inner terms: (27)β‹…(97)=187β‹…7=1849(2 \sqrt{7}) \cdot (9 \sqrt{7}) = 18 \sqrt{7 \cdot 7} = 18 \sqrt{49}.
  4. Last: Multiply the last terms: (27)β‹…(βˆ’56)=βˆ’107β‹…6=βˆ’1042(2 \sqrt{7}) \cdot (-5 \sqrt{6}) = -10 \sqrt{7 \cdot 6} = -10 \sqrt{42}.

Now, we combine these results:

8142βˆ’4536+1849βˆ’104281 \sqrt{42} - 45 \sqrt{36} + 18 \sqrt{49} - 10 \sqrt{42}

Simplifying the Terms

Now that we have expanded the expression, the next step is to simplify each term individually. This involves simplifying the radicals and performing any possible arithmetic operations. Let's examine each term:

  1. 814281 \sqrt{42}: The number 42 has prime factors 2, 3, and 7. Since there are no perfect square factors, 42\sqrt{42} is already in its simplest form. Thus, this term remains as 814281 \sqrt{42}.
  2. βˆ’4536-45 \sqrt{36}: The square root of 36 is 6, so this term simplifies to βˆ’45β‹…6=βˆ’270-45 \cdot 6 = -270.
  3. 184918 \sqrt{49}: The square root of 49 is 7, so this term simplifies to 18β‹…7=12618 \cdot 7 = 126.
  4. βˆ’1042-10 \sqrt{42}: Similar to the first term, 42\sqrt{42} is already in its simplest form, so this term remains as βˆ’1042-10 \sqrt{42}.

Substituting these simplified terms back into the expression, we have:

8142βˆ’270+126βˆ’104281 \sqrt{42} - 270 + 126 - 10 \sqrt{42}

Combining Like Terms

The final step in simplifying the expression is to combine any like terms. In this case, we have two terms with 42\sqrt{42} and two constant terms. Combining these, we get:

(8142βˆ’1042)+(βˆ’270+126)(81 \sqrt{42} - 10 \sqrt{42}) + (-270 + 126)

Simplifying the radical terms:

714271 \sqrt{42}

Simplifying the constant terms:

βˆ’144-144

Combining these results, the fully simplified expression is:

7142βˆ’14471 \sqrt{42} - 144

This is the final simplified form of the given expression. The process involved expanding the expression using the distributive property, simplifying individual terms by evaluating square roots and combining like terms.

Common Mistakes to Avoid

When simplifying expressions with radicals, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One frequent error is incorrectly combining radicals that are not like terms. For example, attempting to add or subtract 2\sqrt{2} and 3\sqrt{3} directly is incorrect because they do not have the same radicand. Another mistake is failing to simplify radicals completely. Always check if the radicand has any perfect square factors that can be extracted. For instance, 8\sqrt{8} should be simplified to 222\sqrt{2}. A third common error occurs when applying the distributive property. Make sure to multiply each term in the first parenthesis by each term in the second parenthesis, without missing any combinations. Additionally, sign errors are common, especially when dealing with negative terms. Pay close attention to the signs when multiplying and combining terms. Lastly, remember that the square root of a product is the product of the square roots, but this rule does not apply to sums or differences. That is, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}, but a+b≠a+b\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}. By being mindful of these common mistakes and practicing diligently, you can improve your accuracy and confidence in simplifying radical expressions.

Practice Problems

To solidify your understanding of multiplying and simplifying radical expressions, it’s essential to practice with a variety of problems. Here are a few practice problems to get you started:

  1. (35+22)(45βˆ’2)(3 \sqrt{5} + 2 \sqrt{2})(4 \sqrt{5} - \sqrt{2})
  2. (73βˆ’10)(23+310)(7 \sqrt{3} - \sqrt{10})(2 \sqrt{3} + 3 \sqrt{10})
  3. (6+47)(6βˆ’7)(\sqrt{6} + 4 \sqrt{7})(\sqrt{6} - \sqrt{7})
  4. (52βˆ’38)(2+8)(5 \sqrt{2} - 3 \sqrt{8})( \sqrt{2} + \sqrt{8})

For each problem, follow the same steps we used in the example: expand the expression using the distributive property (FOIL), simplify each term by evaluating square roots, and combine like terms. Remember to look for perfect square factors in the radicands and simplify the radicals as much as possible. Working through these problems will help you become more comfortable with the process and develop your skills in simplifying radical expressions. Practice is key to mastering this topic, so don't hesitate to work through additional examples and challenge yourself with more complex problems. As you gain experience, you'll become more efficient and accurate in your calculations. By consistently practicing and applying the principles discussed in this guide, you'll build a strong foundation in simplifying radical expressions.

Conclusion

Simplifying expressions with radicals requires a combination of algebraic skills and a solid understanding of radical properties. By following a systematic approach, such as the distributive property and combining like terms, complex expressions can be simplified effectively. In this article, we meticulously worked through the simplification of the expression (96+27)(97βˆ’56)(9 \sqrt{6}+2 \sqrt{7})(9 \sqrt{7}-5 \sqrt{6}), providing detailed steps and explanations. We began by understanding the basics of radicals, then moved on to the step-by-step multiplication process, followed by simplifying individual terms, and finally combining like terms to arrive at the simplified form: 7142βˆ’14471 \sqrt{42} - 144. We also highlighted common mistakes to avoid and provided practice problems to reinforce your understanding. Mastering these techniques not only enhances your mathematical proficiency but also builds a strong foundation for more advanced topics in algebra and calculus. Remember, consistent practice and a clear understanding of the underlying principles are the keys to success in simplifying radical expressions. With dedication and effort, you can confidently tackle even the most challenging problems involving radicals. This skill is valuable in various mathematical contexts, making it an essential tool in your mathematical arsenal.