Simplifying $9\sqrt{5}+7\sqrt{5}+4\sqrt{7}$ A Step-by-Step Guide

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. Radicals, often represented by the square root symbol (√), can sometimes appear daunting, but with a solid understanding of the underlying principles, these expressions can be tamed. This article will delve into the process of simplifying the expression $9\sqrt{5} + 7\sqrt{5} + 4\sqrt{7}$, providing a step-by-step guide and shedding light on the core concepts involved.

Understanding Radicals

Before we dive into the specifics of our expression, let's first establish a firm grasp of what radicals are and how they behave. A radical, in its simplest form, represents the root of a number. The most common type of radical is the square root, which asks the question: "What number, when multiplied by itself, equals the number under the radical sign?" For instance, $\sqrt{9}$ asks, "What number multiplied by itself equals 9?" The answer, of course, is 3, since 3 * 3 = 9. Similarly, the cube root (represented by $\sqrt[3]{}$) asks for a number that, when multiplied by itself three times, equals the number under the radical. And so on for higher roots.

The number under the radical sign is called the radicand. In the expression $\sqrt{5}$, 5 is the radicand. Radicals can involve various types of numbers, including integers, fractions, and even variables. When simplifying radical expressions, our goal is to reduce the radicand to its simplest form, often by factoring out perfect squares (for square roots), perfect cubes (for cube roots), or higher powers.

Identifying Like Radicals

The key to simplifying expressions like $9\sqrt{5} + 7\sqrt{5} + 4\sqrt{7}$ lies in recognizing like radicals. Like radicals are terms that have the same radicand and the same index (the small number indicating the type of root, such as the 2 in a square root or the 3 in a cube root). For example, $3\sqrt{2}$ and $5\sqrt{2}$ are like radicals because they both have the same radicand (2) and the same index (2, since they are square roots). However, $3\sqrt{2}$ and $3\sqrt{3}$ are not like radicals because they have different radicands, even though they are both square roots. Similarly, $3\sqrt{2}$ and $3\sqrt[3]{2}$ are not like radicals because they have different indices.

Like radicals can be combined in a similar way to how we combine like terms in algebraic expressions. We can add or subtract the coefficients (the numbers in front of the radical) of like radicals while keeping the radical part the same. This is analogous to how we combine terms like 3x and 5x to get 8x. The 'x' acts as a placeholder, and we simply add the coefficients. In the case of like radicals, the radical acts as the placeholder.

Step-by-Step Simplification of $9\sqrt{5} + 7\sqrt{5} + 4\sqrt{7}$

Now, let's apply these concepts to simplify the given expression: $9\sqrt{5} + 7\sqrt{5} + 4\sqrt{7}$.

1. Identify like radicals: In this expression, we have two terms with $\sqrt{5}$ as the radical: $9\sqrt{5}$ and $7\sqrt{5}$. The term $4\sqrt{7}$ has a different radicand (7) and is therefore not a like radical with the other two terms.

2. Combine like radicals: We can combine $9\sqrt{5}$ and $7\sqrt{5}$ by adding their coefficients: 9 + 7 = 16. This gives us $16\sqrt{5}$.

3. Write the simplified expression: The term $4\sqrt{7}$ cannot be combined with $16\sqrt{5}$ because they are not like radicals. Therefore, the simplified expression is $16\sqrt{5} + 4\sqrt{7}$.

Why Can't We Combine Unlike Radicals?

It's crucial to understand why we can only combine like radicals. Consider the analogy with algebraic terms. We can combine 3x + 5x to get 8x because the 'x' represents the same quantity in both terms. However, we cannot combine 3x + 5y because 'x' and 'y' represent different quantities. Similarly, $\sqrt{5}$ and $\sqrt{7}$ represent different numerical values. $\sqrt{5}$ is the number that, when multiplied by itself, equals 5, while $\sqrt{7}$ is the number that, when multiplied by itself, equals 7. These are distinct numbers, and we cannot simply add their coefficients to combine them.

Imagine trying to add apples and oranges. You can't simply say you have 8 "apple-oranges." You have to keep them separate as apples and oranges. The same principle applies to unlike radicals. We must maintain the distinction between the different radicals in our expression.

Further Examples and Practice

To solidify your understanding, let's look at a few more examples:

• Simplify $5\sqrt{3} - 2\sqrt{3} + \sqrt{2}$.

  • We can combine $5\sqrt{3}$ and $-2\sqrt{3}$ because they are like radicals. 5 - 2 = 3, so we have $3\sqrt{3}$.
  • $\sqrt{2}$ is not a like radical, so it remains separate.
  • The simplified expression is $3\sqrt{3} + \sqrt{2}$.

• Simplify $2\sqrt{12} + \sqrt{27}$.

  • First, we need to simplify the radicals by factoring out perfect squares. $\sqrt{12} = \sqrt{4 * 3} = \sqrt{4} * \sqrt{3} = 2\sqrt{3}$. $\sqrt{27} = \sqrt{9 * 3} = \sqrt{9} * \sqrt{3} = 3\sqrt{3}$.
  • Now we have $2(2\sqrt{3}) + 3\sqrt{3} = 4\sqrt{3} + 3\sqrt{3}$.
  • Combine like radicals: 4 + 3 = 7, so the simplified expression is $7\sqrt{3}$.

Conclusion

Simplifying radical expressions is a fundamental skill in algebra and beyond. By understanding the concept of like radicals and how to combine them, we can effectively reduce complex expressions to their simplest forms. The expression $9\sqrt{5} + 7\sqrt{5} + 4\sqrt{7}$ serves as a clear example of this process, highlighting the importance of identifying like radicals and applying the rules of addition. Remember to always look for opportunities to simplify radicals further by factoring out perfect squares or other powers. With practice, you'll become proficient in handling radical expressions with confidence.

This exploration into simplifying radical expressions not only enhances our mathematical abilities but also sharpens our analytical thinking. The ability to break down complex problems into manageable steps and apply fundamental principles is a valuable skill that extends far beyond the realm of mathematics. So, embrace the challenge of simplifying radicals, and you'll find yourself not only mastering algebra but also developing a more robust problem-solving mindset.

Are you grappling with simplifying mathematical expressions involving radicals? Look no further! This article provides a comprehensive guide to simplifying the expression $9√5 + 7√5 + 4√7$. We will break down the steps involved, explain the underlying concepts, and offer practical tips to master this skill. Radicals, often perceived as complex, become manageable with a clear understanding of the rules and principles governing their manipulation. Let's embark on this journey of simplifying radicals and enhancing your mathematical prowess.

What are Radicals? A Foundation for Simplification

Before we delve into the specifics of our target expression, it's essential to establish a firm grasp of radicals. A radical is a mathematical expression that uses a root, such as a square root, cube root, or nth root. The most common radical is the square root, denoted by the symbol √. The square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 9 (√9) is 3 because 3 * 3 = 9. Similarly, the square root of 25 (√25) is 5 because 5 * 5 = 25. Understanding this fundamental definition is the cornerstone for simplifying more complex radical expressions.

Beyond square roots, we encounter cube roots (∛), fourth roots (∜), and so on. The index of the radical indicates the type of root. In the square root, the index is implicitly 2. In the cube root, the index is 3, meaning we seek a number that, when multiplied by itself three times, yields the original number. For example, the cube root of 8 (∛8) is 2 because 2 * 2 * 2 = 8. The number under the radical symbol is termed the radicand. Simplifying radicals often involves reducing the radicand to its simplest form by factoring out perfect squares, perfect cubes, or higher powers, depending on the index of the radical. This process is analogous to simplifying fractions by reducing them to their lowest terms.

Identifying and Combining Like Radicals: The Key to Simplification

The linchpin of simplifying expressions like $9√5 + 7√5 + 4√7$ is the ability to identify and combine like radicals. Like radicals are radical terms that possess the same radicand and the same index. This means that both the number under the radical symbol and the type of root must be identical for radicals to be considered “like.” For example, $2√3$ and $5√3$ are like radicals because they both have a radicand of 3 and an index of 2 (square root). However, $2√3$ and $2√5$ are not like radicals because they have different radicands, even though they are both square roots. Similarly, $2√3$ and $2∛3$ are not like radicals because they have different indices, despite sharing the same radicand.

The process of combining like radicals mirrors the combination of like terms in algebraic expressions. Just as we can add 3x + 5x to obtain 8x, we can add $2√3 + 5√3$ to get $7√3$. The radical portion acts as a common unit, similar to the variable 'x'. We simply add or subtract the coefficients (the numbers in front of the radical) while preserving the radical itself. This principle is crucial for simplifying radical expressions efficiently and accurately. Failing to recognize and combine like radicals will result in an incomplete simplification.

Step-by-Step Simplification of $9√5 + 7√5 + 4√7$: A Practical Approach

Now, let's apply our understanding of radicals and like radicals to simplify the expression $9√5 + 7√5 + 4√7$. We will follow a systematic, step-by-step approach to ensure clarity and accuracy.

1. Identify Like Radicals: In the expression $9√5 + 7√5 + 4√7$, we observe two terms with the same radicand (5) and the same index (2, as they are square roots): $9√5$ and $7√5$. The term $4√7$ has a different radicand (7) and, therefore, is not a like radical with the other two terms. This initial identification is crucial for the subsequent steps.

2. Combine Like Radicals: We can combine the like radicals $9√5$ and $7√5$ by adding their coefficients. The coefficients are the numbers in front of the radical symbol: 9 and 7. Adding these coefficients, we get 9 + 7 = 16. Therefore, $9√5 + 7√5 = 16√5$. This step leverages the principle of combining like radicals discussed earlier.

3. Write the Simplified Expression: The term $4√7$ cannot be combined with $16√5$ because they are not like radicals. As such, it remains separate in the simplified expression. The final simplified expression is $16√5 + 4√7$. This is the most reduced form of the original expression, as we have combined all possible like terms.

By following these steps, we have successfully simplified the expression $9√5 + 7√5 + 4√7$. The process underscores the importance of identifying like radicals and applying the principle of combining them by adding their coefficients. Remember, the key to accurate simplification lies in a systematic approach and a solid understanding of the underlying concepts.

Why Can't We Combine Unlike Radicals? A Conceptual Understanding

A common point of confusion for students is why we cannot combine unlike radicals. To understand this, let's draw an analogy to algebraic terms. In algebra, we can combine 3x + 5x to get 8x because 'x' represents the same quantity in both terms. However, we cannot combine 3x + 5y because 'x' and 'y' represent different quantities. We treat 'x' and 'y' as distinct entities, and the same principle applies to radicals.

Similarly, $√5$ and $√7$ represent different numerical values. $√5$ is the number that, when multiplied by itself, equals 5, while $√7$ is the number that, when multiplied by itself, equals 7. These are distinct numbers, and we cannot simply add their coefficients to combine them into a single term. Attempting to do so would be mathematically incorrect, akin to adding apples and oranges and claiming to have a single type of fruit.

Imagine visualizing $√5$ and $√7$ as lengths of line segments. If you have 9 segments of length $√5$ and 7 segments of length $√5$, you can combine them to have 16 segments of length $√5$. However, adding 4 segments of length $√7$ is fundamentally different; you cannot combine these segments with the $√5$ segments to create a single, unified length. This conceptual understanding reinforces why we can only combine radicals with identical radicands and indices.

Practice Problems and Advanced Simplification Techniques

To solidify your understanding and enhance your skills in simplifying radical expressions, it's crucial to practice with a variety of problems. Here are a few examples to get you started:

• Simplify $3√2 - 5√2 + 2√3$.
• Simplify $4√18 + √8 - 3√32$.
• Simplify $√27 + 2√12 - √75$.

These problems require you to identify like radicals and combine them. Additionally, some problems, like the second and third examples, require you to simplify the radicals before identifying like terms. This involves factoring out perfect squares from the radicand. For instance, in $√18$, you can factor out 9 (a perfect square) to get $√(9 * 2) = √9 * √2 = 3√2$. This process of simplifying individual radicals often uncovers hidden like radicals that can then be combined.

Advanced simplification techniques may involve dealing with radicals in the denominator of a fraction, a process called rationalizing the denominator. This technique eliminates radicals from the denominator, making the expression easier to work with. It typically involves multiplying both the numerator and the denominator by a suitable radical expression.

Conclusion: Mastering Radical Simplification

Simplifying radical expressions is a fundamental skill in algebra and precalculus. By grasping the concepts of radicals, like radicals, and the principles of combining them, you can effectively tackle a wide range of mathematical problems. The expression $9√5 + 7√5 + 4√7$ serves as a clear illustration of this process, highlighting the importance of a systematic approach and a solid understanding of the underlying concepts.

Remember, practice is paramount to mastering this skill. Work through various examples, focusing on identifying like radicals, simplifying individual radicals, and applying the rules of combination. As you gain proficiency, you'll find that simplifying radicals becomes a natural and intuitive process, empowering you to excel in your mathematical endeavors. So, embrace the challenge, practice diligently, and unlock the power of radical simplification!