Simplify The Square Root Of 84

Simplifying the Square Root of 84: A Comprehensive Guide

Simplifying square roots is a fundamental concept in mathematics, crucial for algebra, calculus, and various other advanced mathematical fields. Understanding how to simplify radicals, particularly a seemingly straightforward one like the square root of 84, lays the groundwork for more complex mathematical operations. This guide delves into the process of simplifying √84, explaining the steps involved and providing additional examples to solidify your understanding.

Understanding Square Roots and Simplification

Before we tackle √84, let's refresh our understanding of square roots and simplification. A 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 x 3 = 9. However, not all square roots result in whole numbers. This is where simplification comes in. Simplifying a square root means expressing it in its simplest form, removing any perfect square factors from within the radical symbol (√).

What is a Perfect Square?

A perfect square is a number that can be obtained by squaring an integer. Examples include:

• 1 (1 x 1)
• 4 (2 x 2)
• 9 (3 x 3)
• 16 (4 x 4)
• 25 (5 x 5)
• and so on...

Identifying perfect squares is crucial for simplifying square roots efficiently.

Step-by-Step Simplification of √84

To simplify √84, we need to find the largest perfect square that divides evenly into 84. Let's break down the process step-by-step:

Step 1: Prime Factorization

The first step is to find the prime factorization of 84. Prime factorization involves expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves).

84 can be factored as:

84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7 = 2² x 3 x 7

Step 2: Identifying Perfect Squares

Now, examine the prime factorization (2² x 3 x 7). We identify that 2² is a perfect square (2 x 2 = 4).

Step 3: Applying the Product Rule of Radicals

The product rule of radicals states that √(a x b) = √a x √b. We can use this rule to separate the perfect square from the remaining factors:

√84 = √(2² x 3 x 7) = √2² x √(3 x 7)

Step 4: Simplifying the Perfect Square

Since √2² = 2, we can simplify the expression further:

√84 = 2√(3 x 7) = 2√21

Therefore, the simplified form of √84 is 2√21.

Further Examples and Practice

Let's explore a few more examples to solidify your understanding of simplifying square roots:

Example 1: Simplifying √147

1. Prime Factorization: 147 = 3 x 49 = 3 x 7²
2. Identifying Perfect Squares: 7² is a perfect square.
3. Applying the Product Rule: √147 = √(3 x 7²) = √3 x √7²
4. Simplifying: √147 = 7√3

Example 2: Simplifying √200

1. Prime Factorization: 200 = 2 x 100 = 2 x 10² = 2 x (2 x 5)² = 2 x 2² x 5² = 2³ x 5²
2. Identifying Perfect Squares: 2² and 5² are perfect squares.
3. Applying the Product Rule: √200 = √(2³ x 5²) = √(2² x 2 x 5²) = √2² x √5² x √2
4. Simplifying: √200 = 2 x 5 x √2 = 10√2

Example 3: Simplifying √128

1. Prime Factorization: 128 = 2 x 64 = 2 x 8² = 2 x (2³) ² = 2 x 2⁶ = 2⁷
2. Identifying Perfect Squares: 2⁶ is a perfect square (2⁶ = 64)
3. Applying the Product Rule: √128 = √(2⁶ x 2) = √2⁶ x √2
4. Simplifying: √128 = 2³√2 = 8√2

Example 4: A More Challenging Example - Simplifying √756

1. Prime Factorization: 756 = 2 x 378 = 2 x 2 x 189 = 2² x 3 x 63 = 2² x 3 x 3 x 21 = 2² x 3² x 3 x 7 = 2² x 3³ x 7
2. Identifying Perfect Squares: 2² and 3² are perfect squares.
3. Applying the Product Rule: √756 = √(2² x 3² x 3 x 7) = √2² x √3² x √(3 x 7)
4. Simplifying: √756 = 2 x 3 x √21 = 6√21

Common Mistakes to Avoid

When simplifying square roots, several common mistakes can lead to incorrect answers. Let's highlight them:

• Incomplete Prime Factorization: Failing to completely factor the number into its prime components can lead to leaving perfect squares within the radical, resulting in an unsimplified answer. Always ensure you've fully broken down the number.

• Incorrect Application of the Product Rule: Misunderstanding or misapplying the product rule of radicals can lead to errors in separating perfect squares. Double-check your steps to ensure accurate application.

• Ignoring Negative Numbers: While the principal square root of a positive number is always positive, it's important to remember that negative numbers also have square roots. The square root of -9 is not a real number but a complex number denoted as 3i.

Applications of Simplifying Square Roots

Simplifying square roots is not just a theoretical exercise. It has practical applications across various mathematical fields and real-world scenarios:

• Geometry: Calculating areas, volumes, and lengths often involves square roots. Simplifying them makes calculations more manageable and efficient.

• Physics: Many physical phenomena involve square roots in their formulas, such as calculating speed, velocity, and acceleration.

• Algebra: Simplifying square roots is crucial for solving equations, simplifying expressions, and working with quadratic equations.

• Calculus: Derivatives and integrals frequently involve manipulating expressions containing square roots.

Conclusion

Simplifying square roots, even seemingly simple ones like √84, is a fundamental skill with far-reaching implications. By mastering the process of prime factorization, identifying perfect squares, and applying the product rule of radicals, you can confidently simplify any square root and unlock a deeper understanding of various mathematical concepts. Remember to practice regularly to solidify your understanding and avoid common mistakes. With consistent practice, simplifying square roots will become second nature, enabling you to tackle more complex mathematical problems with ease.