What Is The Simplified Form Of The Expression 2√18 + 3√2 + √182?

#h1 Decoding the Expression: 2√18 + 3√2 + √182

In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to represent complex mathematical statements in a more concise and manageable form. This article delves into the intricacies of simplifying the expression 2√18 + 3√2 + √182, providing a step-by-step guide to unravel its hidden form. We'll explore the underlying principles of simplifying radicals, unravel the expression's components, and ultimately arrive at its simplified representation. This exploration will not only enhance your understanding of mathematical simplification but also equip you with the tools to tackle similar problems with confidence.

#h2 Laying the Foundation: Understanding Radicals and Simplification

Before we embark on the simplification journey, let's solidify our understanding of the core concepts involved: radicals and simplification. Radicals, denoted by the symbol √, represent the root of a number. For instance, √9 represents the square root of 9, which is 3, since 3 multiplied by itself equals 9. Radicals can involve various roots, such as square roots, cube roots, and so on. Simplification, on the other hand, is the process of transforming a mathematical expression into a simpler, more concise form without altering its inherent value. In the context of radicals, simplification often involves extracting perfect squares (or perfect cubes, etc.) from under the radical sign.

The essence of simplifying radicals lies in identifying and extracting perfect square factors from the radicand, which is the number under the radical sign. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). By factoring out perfect squares, we can rewrite the radical in a simplified form. For instance, √20 can be simplified as follows:

1. Factor the radicand: 20 can be factored as 4 × 5, where 4 is a perfect square.
2. Rewrite the radical: √20 = √(4 × 5)
3. Apply the product property of radicals: √(4 × 5) = √4 × √5
4. Simplify the perfect square: √4 = 2
5. Final simplified form: 2√5

This process of extracting perfect squares allows us to express radicals in their simplest form, making them easier to work with in mathematical operations and calculations. Understanding this fundamental principle is crucial for simplifying the expression at hand.

#h2 Deconstructing the Expression: 2√18 + 3√2 + √182

Now that we have a firm grasp of the principles of radical simplification, let's turn our attention to the expression 2√18 + 3√2 + √182. This expression comprises three terms, each involving a radical. To simplify the expression, we will focus on simplifying each term individually and then combining like terms.

The first term, 2√18, involves the radical √18. To simplify this term, we need to identify any perfect square factors within 18. We can factor 18 as 9 × 2, where 9 is a perfect square (3 × 3). Applying the same steps as before:

1. Rewrite the radical: 2√18 = 2√(9 × 2)
2. Apply the product property of radicals: 2√(9 × 2) = 2(√9 × √2)
3. Simplify the perfect square: √9 = 3
4. Simplified form: 2(3√2) = 6√2

Thus, the first term, 2√18, simplifies to 6√2. This simplification is a crucial step in our overall quest to simplify the entire expression.

The second term, 3√2, is already in its simplest form, as the radicand 2 does not have any perfect square factors other than 1. This term will remain unchanged during the simplification process. Sometimes, the simplest steps are the most crucial, as they maintain the integrity of the expression while we focus on other areas.

The third term, √182, presents a slightly greater challenge. To simplify this term, we need to determine the prime factorization of 182. The prime factorization of 182 is 2 × 7 × 13. Unfortunately, none of these prime factors appear twice, indicating that 182 does not have any perfect square factors other than 1. Therefore, √182 remains in its simplest form. This outcome underscores the importance of thoroughly examining the radicand for potential simplifications.

#h2 The Simplification Journey: Step-by-Step Breakdown

Having deconstructed the expression into its individual terms, we now embark on the journey of simplifying each term and combining them to arrive at the final simplified form. This process involves a series of methodical steps, each building upon the previous one. Let's retrace our steps and consolidate the simplifications we've performed:

1. Simplify 2√18: As we previously determined, 2√18 simplifies to 6√2.
2. Retain 3√2: This term is already in its simplest form and remains unchanged.
3. Acknowledge √182: This term also remains in its simplest form as √182.

Now, we can rewrite the original expression with the simplified terms:

2√18 + 3√2 + √182 = 6√2 + 3√2 + √182

Combining Like Terms:

The next crucial step is to identify and combine like terms. In this context, like terms are those that have the same radical component. In our expression, 6√2 and 3√2 are like terms because they both involve the radical √2. To combine like terms, we simply add their coefficients (the numbers in front of the radical):

6√2 + 3√2 = (6 + 3)√2 = 9√2

The term √182, however, cannot be combined with the other terms because it involves a different radical. It remains separate in the simplified expression.

#h2 The Grand Finale: Arriving at the Simplified Form

After simplifying each term and combining like terms, we arrive at the final simplified form of the expression:

2√18 + 3√2 + √182 = 9√2 + √182

This is the most concise and simplified representation of the original expression. We have successfully untangled the complexities of the initial expression and presented it in a clear and manageable form. This process highlights the power of mathematical simplification in making complex expressions more accessible and understandable.

#h2 Navigating the Multiple-Choice Landscape: Identifying the Correct Answer

Now that we have successfully simplified the expression, let's consider the multiple-choice options provided and identify the correct answer. The options presented are:

A. 6√2 B. 18√2 C. 30√2 D. 36√2

Comparing our simplified expression, 9√2 + √182, with the multiple-choice options, we can immediately observe that none of the options perfectly match our result. This discrepancy arises because the multiple-choice options only consider the combination of the first two terms, neglecting the third term, √182. Therefore, none of the provided options accurately represent the simplified form of the entire expression. This situation underscores the importance of performing the simplification meticulously and considering all terms in the expression.

#h2 Mastering Simplification: A Toolkit for Success

Simplifying radical expressions is a fundamental skill in mathematics, and mastering it requires a combination of understanding core concepts and practicing problem-solving techniques. Let's recap the essential tools and strategies that we employed in simplifying the expression 2√18 + 3√2 + √182:

• Understanding Radicals: A solid grasp of the definition of radicals and their properties is the foundation for simplification. Knowing that radicals represent roots and understanding how to manipulate them is crucial.
• Factoring Radicands: The ability to factor the radicand and identify perfect square factors is key to extracting them from under the radical sign. This skill allows us to rewrite the radical in a simpler form.
• Product Property of Radicals: Applying the product property of radicals, which states that √(a × b) = √a × √b, enables us to separate factors under the radical and simplify them individually.
• Combining Like Terms: Identifying and combining like terms, those with the same radical component, is essential for arriving at the final simplified form. This step condenses the expression and presents it in its most concise representation.
• Prime Factorization: When dealing with larger radicands, prime factorization can be a powerful tool for identifying perfect square factors. This technique systematically breaks down the radicand into its prime components, making it easier to spot repeated factors.
• Meticulous Approach: Simplifying expressions requires a methodical and step-by-step approach. Each step should be performed carefully and accurately to avoid errors and ensure the final result is correct.
• Practice and Perseverance: Like any mathematical skill, mastering simplification requires consistent practice. The more problems you solve, the more comfortable and confident you will become in applying the techniques and strategies involved. Perseverance is key to overcoming challenges and developing a deep understanding of the process.

By incorporating these tools and strategies into your mathematical arsenal, you will be well-equipped to tackle a wide range of simplification problems involving radicals and other mathematical expressions. The journey of simplification is not just about finding the answer; it's about developing a deeper understanding of mathematical principles and honing your problem-solving skills.

#h1 Conclusion: Embracing the Art of Mathematical Simplification

In conclusion, simplifying mathematical expressions is an art form that involves a combination of technical skill and strategic thinking. The journey of simplifying 2√18 + 3√2 + √182 has highlighted the importance of understanding radicals, factoring radicands, applying the product property of radicals, and combining like terms. While the multiple-choice options provided did not perfectly align with our simplified result, the process of simplification itself has equipped us with valuable insights and skills.

By embracing the principles and techniques discussed in this article, you can confidently approach a wide range of simplification problems and unlock the hidden beauty within complex mathematical expressions. Remember, the goal is not just to find the answer but to cultivate a deeper appreciation for the elegance and power of mathematics.