Simplify Square Root Of 85

Simplifying the Square Root of 85: A Comprehensive Guide

Understanding how to simplify square roots is a fundamental skill in mathematics. This comprehensive guide will walk you through the process of simplifying the square root of 85, explaining the concepts involved and providing a step-by-step approach. We'll explore the prime factorization method, discuss why 85 isn't a perfect square, and delve into the approximation techniques used when dealing with irrational numbers. By the end, you'll have a solid grasp of simplifying square roots and be able to apply these techniques to other numbers.

Introduction: What Does it Mean to Simplify a Square Root?

Simplifying a square root means expressing it in its simplest form. A simplified square root has no perfect square factors under the radical symbol (√). A perfect square is a number that results from squaring an integer (e.g., 4, 9, 16, 25 are perfect squares because they are 2², 3², 4², and 5² respectively). The square root of 85, denoted as √85, isn't a whole number; it's an irrational number, meaning it cannot be expressed as a simple fraction. However, we can simplify it to a more manageable form.

Step-by-Step Simplification of √85 using Prime Factorization

The key to simplifying square roots lies in finding the prime factorization of the number under the radical. Prime factorization means expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves).

1. Find the Prime Factors of 85:

   We start by finding the prime factors of 85. We can begin by dividing 85 by the smallest prime number, 2. Since 85 is an odd number, it's not divisible by 2. The next prime number is 3, and 85 is not divisible by 3. However, 85 is divisible by 5:

   85 = 5 x 17

   Both 5 and 17 are prime numbers. Therefore, the prime factorization of 85 is 5 x 17.

2. Rewrite the Square Root:

   Now, we can rewrite √85 using the prime factorization:

   √85 = √(5 x 17)

3. Simplify (or Determine that Simplification isn't Possible):

   Notice that neither 5 nor 17 is a perfect square. There are no pairs of identical prime factors. This means that √85 cannot be simplified further using the method of extracting perfect square factors. The simplified form of √85 is simply √85. While we can't simplify it to a form without a radical, we have confirmed that it's in its simplest form.

Why √85 Cannot Be Simplified Further: A Deeper Look

The inability to simplify √85 further stems from the fact that its prime factorization contains no repeated factors. The simplification process relies on identifying perfect squares within the radicand (the number under the square root symbol). A perfect square always has an even number of identical prime factors in its prime factorization. For instance:

• √36 = √(2² x 3²) = 2 x 3 = 6 (because we have two 2's and two 3's)
• √100 = √(2² x 5²) = 2 x 5 = 10 (two 2's and two 5's)

Since the prime factorization of 85 (5 x 17) lacks repeated factors, we can't extract any perfect squares. This confirms that √85 is already in its simplest radical form.

Approximating the Value of √85

Although we can't simplify √85 to a simpler radical expression, we can approximate its decimal value. We know that:

• √81 = 9
• √100 = 10

Since 85 lies between 81 and 100, √85 must lie between 9 and 10. We can use a calculator to find a more precise approximation:

√85 ≈ 9.22

Advanced Techniques and Further Exploration

While prime factorization is the most common method for simplifying square roots, other techniques exist, particularly when dealing with more complex expressions. These might involve techniques like:

• Rationalizing the denominator: This is used when a square root appears in the denominator of a fraction. It involves multiplying both the numerator and the denominator by a suitable expression to eliminate the square root from the denominator.
• Using the properties of radicals: These properties allow for simplifying expressions with multiple square roots or combining terms with similar radicals. For example, √a x √b = √(a x b).

Frequently Asked Questions (FAQ)

• Q: Is √85 a rational or irrational number?

  • A: √85 is an irrational number because it cannot be expressed as a fraction of two integers.

• Q: Can I simplify √85 by dividing it by a number?

  • A: You can divide the number inside the square root (85) by its factors during prime factorization, but this helps you simplify only if you find perfect square factors. Dividing the entire square root (√85) by a number changes its value.

• Q: How accurate does my approximation of √85 need to be?

  • A: The required accuracy depends on the context of the problem. For most practical purposes, an approximation to a few decimal places (like 9.22) is sufficient. In more advanced mathematical scenarios higher accuracy might be required.

• Q: Are there any online calculators or tools that can help me simplify square roots?

  • A: Yes, many online calculators are available that can calculate and simplify square roots, giving both the simplified radical form and the decimal approximation. However, understanding the underlying principles is crucial for applying these techniques correctly and confidently.

Conclusion: Mastering Square Root Simplification

Simplifying square roots involves understanding prime factorization and recognizing perfect squares. While √85 cannot be simplified to a form without a radical, we've demonstrated that its simplest form is indeed √85. We've also explored how to approximate its decimal value and discussed some more advanced techniques for dealing with more complex square root expressions. By mastering these concepts, you'll gain a deeper understanding of numbers and enhance your mathematical problem-solving abilities. Remember that practicing regularly is key to developing fluency in simplifying square roots and other mathematical operations. The more you practice, the more comfortable and confident you'll become.