Simplifying 5√12 + 2√75 - 9√108 A Step-by-Step Guide

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In this comprehensive guide, we'll delve into simplifying the given expression: 5√12 + 2√75 - 9√108. This involves breaking down the square roots, identifying common factors, and combining like terms to arrive at the most simplified form. Mastering these techniques is crucial for anyone studying algebra or related mathematical fields.

Understanding Square Root Simplification

Before we dive into the problem, let's recap the basics of simplifying square roots. The core idea is to express the number under the square root as a product of its prime factors. If any prime factor appears twice (or an even number of times), we can take it out of the square root. For instance, √4 = √(2*2) = 2. This principle is the cornerstone of simplifying more complex expressions.

Breaking Down the Radicals

Our first step in simplifying 5√12 + 2√75 - 9√108 is to break down each square root individually:

  1. √12: We can express 12 as 4 * 3, where 4 is a perfect square. Therefore, √12 = √(4 * 3) = √4 * √3 = 2√3.
  2. √75: Similarly, 75 can be written as 25 * 3, with 25 being a perfect square. Thus, √75 = √(25 * 3) = √25 * √3 = 5√3.
  3. √108: The number 108 can be expressed as 36 * 3, where 36 is a perfect square. Hence, √108 = √(36 * 3) = √36 * √3 = 6√3.

Substituting the Simplified Radicals

Now that we've simplified each square root, we substitute these back into the original expression:

5√12 + 2√75 - 9√108 becomes 5(2√3) + 2(5√3) - 9(6√3).

Performing the Multiplication

Next, we perform the multiplication for each term:

  • 5 * (2√3) = 10√3
  • 2 * (5√3) = 10√3
  • 9 * (6√3) = 54√3

Our expression now looks like this: 10√3 + 10√3 - 54√3.

Combining Like Terms

The final step is to combine the like terms. In this case, all terms have √3, so we can add and subtract the coefficients:

10√3 + 10√3 - 54√3 = (10 + 10 - 54)√3

Final Simplified Expression

Performing the arithmetic, we get:

(10 + 10 - 54)√3 = -34√3

Therefore, the simplified form of the expression 5√12 + 2√75 - 9√108 is -34√3.

Detailed Breakdown of the Simplification Process

To further clarify the simplification process, let's delve deeper into each step, providing additional insights and alternative approaches.

Prime Factorization: The Key to Simplification

Prime factorization is a fundamental tool in simplifying square roots. It involves breaking down a number into its prime factors – numbers that are only divisible by 1 and themselves. For example, the prime factorization of 12 is 2 * 2 * 3. This method helps us identify perfect square factors within the square root.

When simplifying square roots, the goal is to find pairs of identical prime factors. Each pair can be taken out of the square root as a single factor. In the case of √12, we have a pair of 2s, so we can take a 2 out, leaving √3 inside.

Exploring Alternative Methods

While prime factorization is a reliable method, there are alternative approaches to simplifying square roots. One such approach is to directly look for perfect square factors. For instance, when simplifying √75, you might immediately recognize that 25 is a perfect square and a factor of 75. This can speed up the simplification process.

Common Mistakes to Avoid

When simplifying expressions with square roots, there are a few common mistakes to watch out for:

  1. Forgetting to multiply coefficients: When taking a factor out of a square root, remember to multiply it with any existing coefficients outside the square root.
  2. Incorrectly combining terms: You can only combine terms that have the same square root. For example, you cannot directly combine 2√3 and 3√2.
  3. Not simplifying completely: Always ensure that the number under the square root has no more perfect square factors. If it does, continue simplifying.

Applications of Square Root Simplification

Simplifying square roots is not just an academic exercise; it has practical applications in various fields, including:

  • Geometry: Calculating lengths and areas often involves square roots.
  • Physics: Many physical formulas contain square roots.
  • Engineering: Square roots are used in structural calculations and other engineering problems.
  • Computer Graphics: Square roots are essential in 3D graphics and animation.

Step-by-Step Solution: 5√12 + 2√75 - 9√108

Let's reiterate the step-by-step solution for simplifying the expression 5√12 + 2√75 - 9√108 to solidify your understanding:

  1. Simplify √12: √12 = √(4 * 3) = √4 * √3 = 2√3
  2. Simplify √75: √75 = √(25 * 3) = √25 * √3 = 5√3
  3. Simplify √108: √108 = √(36 * 3) = √36 * √3 = 6√3
  4. Substitute back into the expression: 5(2√3) + 2(5√3) - 9(6√3)
  5. Multiply: 10√3 + 10√3 - 54√3
  6. Combine like terms: (10 + 10 - 54)√3
  7. Simplify: -34√3

Therefore, 5√12 + 2√75 - 9√108 = -34√3

Conclusion

In conclusion, simplifying expressions with square roots requires a systematic approach. By breaking down the square roots, identifying common factors, and combining like terms, we can arrive at the most simplified form. Mastering these techniques is crucial for success in algebra and related fields. Remember to practice these steps with different expressions to build your proficiency and confidence in handling square root simplification problems. The expression 5√12 + 2√75 - 9√108 simplifies to -34√3, a clear demonstration of the power of these simplification techniques.

By understanding and applying the principles outlined in this guide, you'll be well-equipped to tackle a wide range of algebraic problems involving square roots. Remember, practice is key to mastery, so keep working on these types of problems to enhance your skills and understanding. Whether you're a student learning algebra or someone looking to brush up on your math skills, the ability to simplify square roots is a valuable asset.

This guide has provided a comprehensive explanation of how to simplify the expression 5√12 + 2√75 - 9√108, highlighting the importance of prime factorization, identifying perfect square factors, and combining like terms. With a firm grasp of these concepts, you'll be able to confidently approach similar problems and achieve accurate results. Continue to explore and practice, and you'll find that simplifying square roots becomes a straightforward and rewarding process.