Simplify Radicals A Step-by-Step Guide To 4√98 + 12√2 - √18

Introduction

In the realm of mathematics, simplifying radical expressions is a fundamental skill, crucial for tackling more complex problems in algebra, calculus, and beyond. This article delves into the process of simplifying the expression $4 \sqrt{98} + 12 \sqrt{2} - \sqrt{18}$, providing a step-by-step guide that not only arrives at the solution but also elucidates the underlying principles of radical simplification. Mastering these techniques empowers students and enthusiasts alike to approach similar challenges with confidence and precision. This exploration begins with understanding the nature of radicals and their properties, paving the way for a methodical simplification process that enhances both comprehension and computational proficiency. The journey through this mathematical simplification will highlight the significance of identifying perfect square factors, employing the product property of square roots, and combining like terms. The goal is to transform the given expression into its simplest form, thereby fostering a deeper appreciation for the elegance and efficiency of mathematical operations. This introduction sets the stage for a detailed examination of the steps involved, ensuring that readers grasp not only the how but also the why behind each manipulation.

Understanding Radicals

Before diving into the simplification process, it's crucial to understand radicals. A radical, denoted by the symbol $\sqrt{}$, represents the root of a number. In the context of square roots, we are looking for a number that, when multiplied by itself, equals the number under the radical sign (the radicand). For example, $\sqrt{9}$ is 3 because 3 * 3 = 9. Simplifying radicals involves expressing them in their simplest form, where the radicand has no square factors other than 1. This process often requires breaking down the radicand into its prime factors and identifying pairs of identical factors. These pairs can then be taken out of the radical as single factors, reducing the radicand to its smallest possible value. Understanding this foundational concept is key to tackling more complex expressions involving multiple radicals and coefficients. Moreover, it lays the groundwork for grasping more advanced mathematical concepts such as rationalizing denominators and solving radical equations. By mastering the art of simplifying radicals, one gains a valuable tool for navigating a wide array of mathematical problems with greater ease and accuracy. The significance of this skill extends beyond the classroom, finding applications in various fields that rely on mathematical precision and problem-solving.

Step-by-Step Simplification of $4 \sqrt{98} + 12 \sqrt{2} - \sqrt{18}$

Step 1: Simplify $\sqrt{98}$

The primary step in simplifying the expression $4 \sqrt{98} + 12 \sqrt{2} - \sqrt{18}$ is to simplify each radical term individually. We begin with $\sqrt{98}$. To simplify $\sqrt{98}$, we need to find the largest perfect square that divides 98. The perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. We can see that 49 is the largest perfect square that divides 98 (98 = 49 * 2). Therefore, we can rewrite $\sqrt{98}$ as $\sqrt{49 * 2}$. Using the property of square roots that $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we get $\sqrt{49 * 2} = \sqrt{49} * \sqrt{2}$. Since $\sqrt{49} = 7$, we have $\sqrt{98} = 7\sqrt{2}$. This simplification is crucial as it transforms the radical into a form that can be more easily combined with other terms in the expression. The ability to identify and extract perfect square factors is a cornerstone of radical simplification, enabling the transformation of complex expressions into more manageable forms. This process not only simplifies the individual radical but also prepares the expression for further simplification by combining like terms.

Step 2: Substitute and Simplify $4 \sqrt{98}$

Now that we've simplified $\sqrt{98}$ to $7\sqrt{2}$, we can substitute this back into the original expression. We have $4 \sqrt{98}$, so replacing $\sqrt{98}$ with $7\sqrt{2}$ gives us $4 * 7\sqrt{2}$. Multiplying the coefficients, we get $28\sqrt{2}$. This step is a straightforward application of substitution, but it's a critical one. It demonstrates how simplifying individual components of an expression can lead to a simpler overall form. By replacing the more complex radical with its simplified equivalent, we reduce the complexity of the term and make it easier to work with in subsequent steps. The result, $28\sqrt{2}$, is a simplified expression that can now be combined with other terms in the original equation. This process of substitution and simplification is a common technique in algebra and is essential for solving a wide range of mathematical problems. The careful execution of this step ensures that the expression is being manipulated correctly and that the final result will be accurate.

Step 3: Simplify $\sqrt{18}$

Next, we simplify $\sqrt{18}$. Similar to step one, we look for the largest perfect square that divides 18. The perfect squares are 1, 4, 9, 16, and so on. We find that 9 is the largest perfect square that divides 18 (18 = 9 * 2). Thus, we can rewrite $\sqrt{18}$ as $\sqrt{9 * 2}$. Again, using the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we get $\sqrt{9 * 2} = \sqrt{9} * \sqrt{2}$. Since $\sqrt{9} = 3$, we have $\sqrt{18} = 3\sqrt{2}$. This simplification is crucial for making the original expression easier to manage. Just as with simplifying $\sqrt{98}$, identifying the perfect square factor in $\sqrt{18}$ allows us to express the radical in a more concise form. The ability to break down a radicand into its factors and extract the square root of the perfect square is a fundamental skill in simplifying radical expressions. This step not only reduces the complexity of the term but also reveals a common radical, $\sqrt{2}$, which will be important for combining like terms later in the simplification process. The methodical approach to simplifying each radical term ensures accuracy and efficiency in solving the problem.

Step 4: Substitute Simplified Radicals into the Original Expression

Now that we've simplified $\sqrt{98}$ to $7\sqrt{2}$ and $\sqrt{18}$ to $3\sqrt{2}$, we can substitute these simplified forms back into the original expression: $4 \sqrt{98} + 12 \sqrt{2} - \sqrt{18}$. Substituting gives us $4(7\sqrt{2}) + 12\sqrt{2} - 3\sqrt{2}$. This substitution is a pivotal step in the simplification process. It replaces the more complex radicals with their simplified equivalents, making the expression easier to manipulate. By substituting the simplified forms, we transform the original problem into an expression with like terms, which can then be combined. This step highlights the power of simplification in mathematics, where complex problems can be made more manageable by breaking them down into smaller, simpler parts. The careful substitution of the simplified radicals is essential for ensuring accuracy in the final result. It sets the stage for the final step of combining like terms to arrive at the simplest form of the expression.

Step 5: Combine Like Terms

After substituting the simplified radicals, our expression is $4(7\sqrt{2}) + 12\sqrt{2} - 3\sqrt{2}$. First, we multiply 4 by $7\sqrt{2}$ to get $28\sqrt{2}$. Now the expression is $28\sqrt{2} + 12\sqrt{2} - 3\sqrt{2}$. We can see that all terms have the same radical, $\sqrt{2}$, which means they are like terms. We can combine these terms by adding and subtracting their coefficients. Adding the coefficients, we have 28 + 12 - 3. 28 + 12 equals 40, and 40 - 3 equals 37. Therefore, the simplified expression is $37\sqrt{2}$. This final step demonstrates the importance of recognizing and combining like terms in algebraic expressions. By treating the radical $\sqrt{2}$ as a common factor, we can simplify the expression by performing arithmetic operations on the coefficients. The result, $37\sqrt{2}$, is the simplest form of the original expression, and it represents the culmination of the simplification process. This step not only provides the final answer but also reinforces the fundamental principles of algebraic manipulation and simplification.

Final Answer

The simplified form of the expression $4 \sqrt{98} + 12 \sqrt{2} - \sqrt{18}$ is $37\sqrt{2}$.

Conclusion

In conclusion, the simplification of radical expressions, as demonstrated with $4 \sqrt{98} + 12 \sqrt{2} - \sqrt{18}$, is a crucial skill in mathematics. This process involves identifying perfect square factors within the radicands, applying the product property of square roots, and combining like terms. Each step, from breaking down $\sqrt{98}$ and $\sqrt{18}$ to substituting and simplifying, plays a vital role in achieving the final simplified form, $37\sqrt{2}$. Mastering these techniques enhances one's ability to tackle more complex algebraic problems and fosters a deeper understanding of mathematical principles. The methodical approach to simplification not only yields the correct answer but also reinforces the importance of precision and attention to detail in mathematical operations. This skill is invaluable for students and professionals alike, enabling them to confidently navigate a wide range of mathematical challenges. The ability to simplify radicals is not just about finding the answer; it's about understanding the underlying structure of mathematical expressions and developing the problem-solving skills necessary for success in mathematics and beyond. This comprehensive guide has hopefully shed light on the process, making it more accessible and understandable for all.