Simplify 2√18 - 5√32 + 7√162 A Step By Step Guide

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Simplifying radical expressions is a fundamental skill in mathematics, particularly in algebra and pre-calculus. This article delves into the step-by-step process of simplifying the given expression: 2√18 - 5√32 + 7√162. We will explore how to break down each radical, identify perfect square factors, and combine like terms to arrive at the simplest form. This process not only enhances understanding of radical operations but also prepares students for more advanced mathematical concepts.

Understanding the Basics of Radical Simplification

Before we tackle the main expression, let's recap the basics of simplifying square roots. The key idea is to express the number under the square root (the radicand) as a product of its factors, with at least one factor being a perfect square. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). This allows us to extract the square root of the perfect square factor and simplify the expression.

For instance, consider √8. We can rewrite 8 as 4 × 2, where 4 is a perfect square. Thus, √8 = √(4 × 2) = √4 × √2 = 2√2. This simplification makes the radical expression easier to work with in further calculations.

This method hinges on the property √(a × b) = √a × √b, which holds true for non-negative numbers a and b. By applying this property, we can break down complex radicals into simpler forms.

Step-by-Step Simplification of 2√18

Let's start by simplifying the first term, 2√18. Our goal is to find the largest perfect square that divides 18. The factors of 18 are 1, 2, 3, 6, 9, and 18. Among these, 9 is the largest perfect square (since 9 = 3²). Therefore, we can rewrite 18 as 9 × 2.

So, 2√18 = 2√(9 × 2). Applying the property of square roots, we get 2√18 = 2(√9 × √2). Since √9 = 3, the expression simplifies to 2(3√2) = 6√2.

By breaking down the radicand and identifying the perfect square factor, we've successfully simplified 2√18 to 6√2. This approach is crucial for simplifying more complex radical expressions as well.

Simplifying -5√32

Next, let's simplify the second term, -5√32. We need to find the largest perfect square that divides 32. The factors of 32 are 1, 2, 4, 8, 16, and 32. Among these, 16 is the largest perfect square (since 16 = 4²). Thus, we can rewrite 32 as 16 × 2.

So, -5√32 = -5√(16 × 2). Applying the property of square roots, we get -5√32 = -5(√16 × √2). Since √16 = 4, the expression simplifies to -5(4√2) = -20√2.

This simplification demonstrates the importance of identifying the largest perfect square factor to minimize the remaining radicand. Simplifying -5√32 to -20√2 makes it easier to combine with other like terms later on.

Simplifying 7√162

Now, let's tackle the third term, 7√162. We need to find the largest perfect square that divides 162. To do this, we can list the factors of 162: 1, 2, 3, 6, 9, 18, 27, 54, 81, and 162. Among these, 81 is the largest perfect square (since 81 = 9²). Therefore, we rewrite 162 as 81 × 2.

So, 7√162 = 7√(81 × 2). Applying the property of square roots, we get 7√162 = 7(√81 × √2). Since √81 = 9, the expression simplifies to 7(9√2) = 63√2.

Simplifying 7√162 to 63√2 is a crucial step in solving the original expression. By identifying and extracting the largest perfect square, we've made the radical expression much simpler to handle.

Combining Like Terms

After simplifying each term individually, we now have the expression in the form 6√2 - 20√2 + 63√2. Notice that all three terms have the same radical part, √2. This means they are like terms, and we can combine them just like we combine algebraic terms with the same variable.

To combine like terms, we simply add or subtract their coefficients (the numbers in front of the radical). In this case, we have 6 - 20 + 63 as the coefficients.

So, 6√2 - 20√2 + 63√2 = (6 - 20 + 63)√2. Performing the arithmetic, we get (6 - 20 + 63) = -14 + 63 = 49. Therefore, the combined expression is 49√2.

This step highlights the importance of simplifying radicals before attempting to combine terms. Only terms with the same radical part can be combined, making simplification a crucial prerequisite.

Final Simplified Expression

After simplifying each radical term and combining like terms, we arrive at the final simplified expression: 49√2.

This result is the simplest form of the original expression, 2√18 - 5√32 + 7√162. It represents the same value but is expressed in a more concise and manageable form. The process we followed involved breaking down each radical into its prime factors, identifying perfect square factors, extracting their square roots, and finally, combining like terms.

Importance of Simplifying Radical Expressions

Simplifying radical expressions is not just an exercise in algebraic manipulation; it has practical applications in various areas of mathematics and science. Simplified radicals are easier to work with in calculations, especially when dealing with fractions or further algebraic operations. They also provide a clearer understanding of the numerical value of the expression.

For example, consider the expression (49√2)/7. If we didn't simplify the original expression to 49√2, we would have to deal with much larger numbers and potentially make errors in the calculation. But now, we can easily simplify (49√2)/7 to 7√2.

Moreover, in fields like physics and engineering, where calculations often involve square roots, simplifying radicals can lead to more accurate and efficient solutions. Therefore, mastering the technique of simplifying radical expressions is a valuable skill for anyone pursuing STEM fields.

Common Mistakes to Avoid

While simplifying radical expressions, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help avoid errors and ensure accuracy.

  1. Failing to Identify the Largest Perfect Square: Sometimes, students may find a perfect square factor but not the largest one. This can lead to incomplete simplification. For example, when simplifying √32, identifying 4 as a perfect square is a good start, but recognizing 16 as the largest perfect square allows for a more direct simplification.
  2. Incorrectly Applying the Distributive Property: When there is a coefficient outside the radical, it's crucial to remember that it only multiplies the simplified radical and not the original radicand. For example, 2√18 simplifies to 2(3√2) = 6√2, not √36.
  3. Combining Unlike Terms: As mentioned earlier, only terms with the same radical part can be combined. A common mistake is to try to add or subtract terms with different radicals (e.g., √2 and √3) as if they were like terms.
  4. Forgetting to Simplify Completely: Always ensure that the radicand has no remaining perfect square factors. If further simplification is possible, the expression is not in its simplest form.

By avoiding these common mistakes and carefully following the steps outlined in this article, you can confidently simplify radical expressions.

Alternative Methods for Simplification

While the method described above is effective, there are alternative approaches to simplifying radical expressions that some students may find helpful. One such method involves prime factorization.

Prime Factorization Method

The prime factorization method involves breaking down the radicand into its prime factors. This can be particularly useful when dealing with large numbers or when the perfect square factors are not immediately obvious. Let's illustrate this method with the expression √162.

  1. Find the Prime Factorization: The prime factorization of 162 is 2 × 3 × 3 × 3 × 3, which can be written as 2 × 3⁴.
  2. Identify Pairs of Factors: For every pair of identical prime factors, one factor can be brought outside the square root. In this case, we have two pairs of 3s (3 × 3), so we can take out 3 × 3 = 9.
  3. Write the Simplified Expression: The remaining factor inside the square root is 2. Thus, √162 = √(2 × 3⁴) = 3²√2 = 9√2.

This method provides a systematic way to find perfect square factors, especially when they are not immediately apparent. It reinforces the concept of prime factorization and its application in simplifying radicals.

Using a Factor Tree

Another visual aid for finding prime factors is the factor tree. A factor tree is a diagram that breaks down a number into its factors, and then each factor into its factors, until all factors are prime numbers. For example, to create a factor tree for 32:

  1. Start with 32 and branch out to two factors, such as 4 and 8.
  2. Continue branching out from each factor until you reach prime numbers. 4 can be broken down into 2 × 2, and 8 can be broken down into 2 × 4, which further breaks down into 2 × 2 × 2.
  3. The prime factorization of 32 is 2 × 2 × 2 × 2 × 2 = 2⁵.

Using the prime factorization, we can rewrite √32 as √(2⁵) = √(2⁴ × 2) = √(16 × 2) = 4√2. This visual method can be particularly helpful for students who are more visual learners.

Practice Problems

To solidify your understanding of simplifying radical expressions, it's essential to practice. Here are a few practice problems with varying levels of difficulty:

  1. Simplify √48
  2. Simplify 3√50 - 2√18
  3. Simplify √75 + 4√12 - 2√27
  4. Simplify 2√24 + 5√54 - √96

Working through these problems will not only reinforce the steps involved in simplifying radicals but also build confidence in your ability to tackle more complex expressions. The answers to these practice problems are provided at the end of this article.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  1. √48 = √(16 × 3) = 4√3
  2. 3√50 - 2√18 = 3√(25 × 2) - 2√(9 × 2) = 3(5√2) - 2(3√2) = 15√2 - 6√2 = 9√2
  3. √75 + 4√12 - 2√27 = √(25 × 3) + 4√(4 × 3) - 2√(9 × 3) = 5√3 + 4(2√3) - 2(3√3) = 5√3 + 8√3 - 6√3 = 7√3
  4. 2√24 + 5√54 - √96 = 2√(4 × 6) + 5√(9 × 6) - √(16 × 6) = 2(2√6) + 5(3√6) - 4√6 = 4√6 + 15√6 - 4√6 = 15√6

By checking your work against these solutions, you can identify any areas where you may need further practice or clarification. Consistent practice is key to mastering the skill of simplifying radical expressions.

Conclusion

Simplifying the expression 2√18 - 5√32 + 7√162 is a comprehensive exercise that encompasses various aspects of radical simplification. By breaking down each term, identifying perfect square factors, and combining like terms, we arrived at the simplified form: 49√2.

This process not only demonstrates the mechanics of radical simplification but also highlights the importance of understanding the underlying principles. By mastering these techniques, students can confidently tackle a wide range of mathematical problems involving radical expressions. The ability to simplify radicals is a foundational skill that paves the way for more advanced mathematical concepts and applications in various fields. Remember, consistent practice and attention to detail are key to success in simplifying radical expressions and beyond.

Therefore, the correct answer is C) 49√2.