Simplify √108 + 4√3 A Step-by-Step Guide
At the heart of algebra lies the crucial skill of simplifying radical expressions. In this comprehensive guide, we embark on a journey to demystify the process of simplifying the expression √108 + 4√3. This seemingly complex expression can be elegantly simplified using fundamental principles of radical manipulation. Mastering these techniques will not only enhance your understanding of algebraic concepts but also equip you with essential tools for tackling a wide range of mathematical problems. We will break down each step, providing clear explanations and insightful tips to ensure you grasp the underlying concepts thoroughly. By the end of this guide, you'll be able to confidently simplify expressions involving radicals, transforming them into their most concise and manageable forms.
Simplifying radicals is an important skill in algebra, often used to rewrite expressions in a more manageable or understandable form. Before diving into the solution, let’s first understand the concept of simplifying radicals. Radicals, often represented by the square root symbol (√), indicate the root of a number. For instance, √9 represents the square root of 9, which is 3. However, when dealing with larger numbers or numbers that are not perfect squares, simplifying radicals becomes necessary. The core idea behind simplifying radicals involves identifying perfect square factors within the radicand (the number under the square root symbol). A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). By extracting these perfect square factors, we can rewrite the radical expression in a simpler form. This process not only makes the expression easier to work with but also provides a clearer representation of the number's value. For example, √32 can be simplified by recognizing that 32 has a perfect square factor of 16 (since 32 = 16 × 2). Therefore, √32 can be rewritten as √(16 × 2), which then simplifies to √16 × √2, or 4√2. This simplification makes it easier to understand the magnitude of the number and to perform further calculations with it. This method is not only applicable to square roots but can also be extended to cube roots and higher-order roots. The principle remains the same: identify perfect power factors within the radicand and extract them to simplify the expression.
To simplify √108 + 4√3, we must first focus on simplifying √108. The key technique here is prime factorization. Prime factorization involves breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. A prime number is a number greater than 1 that has only two distinct divisors: 1 and itself (examples include 2, 3, 5, 7, 11, and so on). By expressing a number as a product of its prime factors, we can easily identify perfect square factors, which are essential for simplifying radicals. For example, let’s consider the number 108. To find its prime factors, we start by dividing it by the smallest prime number, 2, if it's divisible. 108 divided by 2 is 54, so we have 108 = 2 × 54. Now, we continue with 54. 54 is also divisible by 2, giving us 54 = 2 × 27. So, 108 = 2 × 2 × 27. Next, we look at 27. 27 is not divisible by 2, so we move to the next prime number, 3. 27 divided by 3 is 9, thus 27 = 3 × 9. Now, 9 is also divisible by 3, giving us 9 = 3 × 3. Putting it all together, the prime factorization of 108 is 2 × 2 × 3 × 3 × 3, which can also be written as 2² × 3² × 3. The process of prime factorization is fundamental because it allows us to systematically break down any composite number (a number with more than two factors) into its fundamental building blocks. This is particularly useful in simplifying radicals because perfect square factors become readily apparent in the prime factorization. By identifying pairs of identical prime factors, we can extract them from under the square root, thereby simplifying the radical expression. In the case of √108, the prime factorization helps us identify the perfect square factors, paving the way for simplification. This technique is not only valuable in simplifying square roots but also in simplifying cube roots and higher-order roots, where identifying perfect cube factors or perfect nth-power factors is necessary. The ability to perform prime factorization efficiently is a crucial skill in algebra and number theory.
Having determined the prime factorization of 108 as 2² × 3² × 3, we can now simplify √108. The goal is to identify and extract perfect square factors from under the square root symbol. A perfect square is a number that can be obtained by squaring an integer, such as 4 (2²), 9 (3²), 16 (4²), and so on. In the prime factorization of 108, we can see that 2² and 3² are perfect square factors. To simplify √108, we rewrite it as √(2² × 3² × 3). Using the property of square roots that √(a × b) = √a × √b, we can separate the factors under the square root: √108 = √(2² × 3² × 3) = √2² × √3² × √3. Now, we can take the square root of the perfect squares: √2² = 2 and √3² = 3. Substituting these values back into the expression, we get √108 = 2 × 3 × √3. Multiplying the integers, we find that √108 = 6√3. This simplified form of √108 is much easier to work with than the original expression. The process of extracting perfect squares from under the square root is a fundamental technique in simplifying radicals. It allows us to reduce complex radical expressions into simpler, more manageable forms. The key is to identify the perfect square factors within the radicand, rewrite the expression by separating these factors, and then take the square root of the perfect squares. The remaining factors, which are not perfect squares, stay under the square root symbol. This technique is not only applicable to square roots but can also be extended to cube roots and higher-order roots, where the goal is to identify and extract perfect cube factors or perfect nth-power factors. The ability to simplify radicals efficiently is a crucial skill in algebra and calculus, making it an essential tool for solving a wide range of mathematical problems. By mastering this technique, you can significantly enhance your understanding of algebraic concepts and improve your problem-solving abilities.
Now that we have simplified √108 to 6√3, we can substitute this back into the original expression: √108 + 4√3. The expression becomes 6√3 + 4√3. At this point, we are dealing with what are known as “like terms.” Like terms, in the context of radical expressions, are terms that have the same radical part. In this case, both 6√3 and 4√3 have the same radical part, which is √3. To combine like terms, we treat the radical part (√3) as a common factor and simply add the coefficients (the numbers in front of the radical). This is analogous to combining like terms in algebraic expressions, such as 6x + 4x, where we would add the coefficients 6 and 4 to get 10x. Similarly, in the expression 6√3 + 4√3, we add the coefficients 6 and 4. Adding 6 and 4 gives us 10. Therefore, 6√3 + 4√3 = (6 + 4)√3 = 10√3. This process of combining like terms is a fundamental technique in simplifying algebraic expressions, and it applies equally to radical expressions. By identifying and combining like terms, we can reduce complex expressions into simpler, more manageable forms. This not only makes the expressions easier to understand but also facilitates further calculations or manipulations. The concept of like terms extends beyond simple addition and subtraction. It is also relevant when dealing with multiplication and division of radical expressions. In those cases, we can only combine terms that have the same radical part. If the radical parts are different, the terms cannot be combined directly. The ability to recognize and combine like terms is a crucial skill in algebra, enabling you to simplify a wide range of expressions and solve mathematical problems more efficiently. By mastering this technique, you can significantly enhance your algebraic proficiency and improve your problem-solving abilities.
Therefore, the simplified form of √108 + 4√3 is 10√3. This result is obtained by first simplifying √108 to 6√3 through prime factorization and extraction of perfect square factors, and then combining the like terms 6√3 and 4√3. The entire process highlights the importance of several fundamental algebraic techniques, including prime factorization, simplifying radicals, and combining like terms. Each step plays a crucial role in transforming the original expression into its simplest form. Prime factorization allows us to break down numbers into their fundamental building blocks, revealing perfect square factors that can be extracted from under the square root symbol. This technique is not only essential for simplifying radicals but also for various other mathematical operations, such as finding the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers. Simplifying radicals involves identifying and extracting perfect square factors, thereby reducing the expression to its most manageable form. This process makes the expression easier to understand and work with, facilitating further calculations or manipulations. Combining like terms is another crucial technique in algebra, enabling us to simplify expressions by grouping terms that share the same variable or radical part. By adding or subtracting the coefficients of like terms, we can reduce complex expressions into simpler forms. The ability to simplify expressions effectively is a cornerstone of algebraic proficiency. It not only makes mathematical problems easier to solve but also provides a deeper understanding of the underlying concepts. By mastering these techniques, you can significantly enhance your algebraic skills and improve your problem-solving abilities. The simplified form, 10√3, is not only more concise but also provides a clearer representation of the number's value. This simplification makes it easier to compare with other numbers, perform further calculations, and understand the overall magnitude of the expression.
In conclusion, simplifying √108 + 4√3 to 10√3 demonstrates a powerful application of fundamental algebraic principles. The process involves prime factorization, extraction of perfect square factors, and the combination of like terms. These techniques are not only crucial for simplifying radical expressions but also for a wide range of mathematical problems. Mastering these skills will significantly enhance your algebraic proficiency and problem-solving abilities. The journey from the initial expression, √108 + 4√3, to the simplified form, 10√3, showcases the elegance and efficiency of mathematical simplification. Each step is grounded in sound mathematical principles, and the final result provides a clearer and more manageable representation of the original expression. Prime factorization allows us to break down numbers into their fundamental building blocks, revealing hidden structures and patterns. This technique is not only valuable in simplifying radicals but also in various other areas of mathematics, such as number theory and cryptography. Simplifying radicals involves transforming complex expressions into simpler forms, making them easier to understand and work with. This skill is essential for solving equations, performing calculations, and understanding the properties of numbers. Combining like terms is a fundamental technique in algebra, enabling us to group terms that share the same variable or radical part. This process simplifies expressions and makes them easier to manipulate. The ability to simplify expressions effectively is a hallmark of mathematical expertise. It not only makes problem-solving more efficient but also fosters a deeper understanding of the underlying concepts. By mastering these techniques, you can unlock the power of algebra and gain confidence in your mathematical abilities. The simplified form, 10√3, is a testament to the power of simplification. It represents the original expression in its most concise and manageable form, making it easier to compare with other numbers, perform further calculations, and understand the overall magnitude of the expression. This simplification is not just a matter of aesthetics; it is a crucial step in solving mathematical problems and gaining insights into the nature of numbers and expressions. So, embrace the art of simplification, and watch your mathematical skills flourish.
