Simplify Square Root Of 85

Simplifying the Square Root of 85: A Comprehensive Guide

The square root of 85, denoted as √85, is an irrational number. This means it cannot be expressed as a simple fraction and its decimal representation goes on forever without repeating. While we can't find a perfect integer answer, we can simplify √85 to its most concise radical form. This process involves understanding prime factorization and the properties of square roots. This article will guide you through the process step-by-step, explaining the underlying mathematical concepts and addressing common questions.

Understanding Square Roots and Prime Factorization

Before we dive into simplifying √85, let's review some fundamental concepts. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 (√9) is 3 because 3 x 3 = 9.

Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...). Prime factorization helps us identify perfect squares within a number, which is crucial for simplifying square roots.

Step-by-Step Simplification of √85

1. Find the Prime Factorization of 85: We start by finding the prime factors of 85. We can see that 85 is divisible by 5:

   85 = 5 x 17

   Both 5 and 17 are prime numbers, so we have our prime factorization.

2. Identify Perfect Squares (if any): Looking at the prime factorization (5 x 17), we don't find any perfect squares. A perfect square is a number that can be obtained by squaring an integer (e.g., 4 = 2², 9 = 3², 16 = 4²). Neither 5 nor 17 is a perfect square.

3. Simplify the Square Root: Since there are no perfect square factors within the prime factorization of 85, we cannot simplify √85 further. It remains in its simplest radical form as √85.

Approximating √85

Although we can't simplify √85 to a simpler radical form, we can approximate its value. We know that √81 = 9 and √100 = 10. Since 85 lies between 81 and 100, √85 must lie between 9 and 10.

To get a more precise approximation, we can use a calculator:

√85 ≈ 9.22

Why √85 Cannot Be Simplified Further: A Deeper Look

The inability to simplify √85 further stems from the fact that its prime factorization contains only distinct prime numbers (5 and 17), each appearing only once. The simplification of square roots relies on extracting perfect squares. For instance, let's look at an example where simplification is possible:

Simplify √72:

1. Prime Factorization: 72 = 2 x 2 x 2 x 3 x 3 = 2² x 2 x 3²

2. Identify Perfect Squares: We have 2² and 3²

3. Simplify: √72 = √(2² x 3² x 2) = √2² x √3² x √2 = 2 x 3 x √2 = 6√2

In this case, we were able to extract the perfect squares (2² and 3²) and simplify the expression. However, in the case of √85, no such perfect squares exist within its prime factorization.

Working with Square Roots in Equations

Understanding how to simplify square roots is essential when solving equations involving square roots. Consider the following equation:

x² = 85

To solve for x, we take the square root of both sides:

√x² = ±√85

x = ±√85

The ± sign indicates that there are two solutions: x = √85 and x = -√85. Again, we cannot simplify √85 further, leaving the solution in its simplest radical form.

Frequently Asked Questions (FAQ)

Q1: Is √85 a rational or irrational number?

A1: √85 is an irrational number. Irrational numbers cannot be expressed as a fraction of two integers and have non-repeating, non-terminating decimal expansions.

Q2: Can I simplify √85 any further than √85?

A2: No. The prime factorization of 85 (5 x 17) contains no perfect squares, making further simplification impossible.

Q3: How do I estimate the value of √85 without a calculator?

A3: You can estimate by considering perfect squares close to 85. Since 9² = 81 and 10² = 100, √85 lies between 9 and 10. You can refine the estimate by considering how close 85 is to 81 or 100.

Q4: What if I had a number like √180? How would I simplify that?

A4: Let's simplify √180:

1. Prime Factorization: 180 = 2² x 3² x 5

2. Identify Perfect Squares: We have 2² and 3²

3. Simplify: √180 = √(2² x 3² x 5) = √2² x √3² x √5 = 2 x 3 x √5 = 6√5

Q5: Are there any tricks to quickly determine if a number has a perfect square factor?

A5: While there's no single "trick," familiarity with perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on) and their multiples is helpful. Looking for divisibility by these numbers is a good starting point. Also, understanding the prime factorization helps significantly.

Conclusion

Simplifying square roots involves finding the prime factorization of the number under the radical sign and identifying any perfect square factors. While √85 cannot be simplified beyond its simplest form (√85), understanding the process allows you to handle more complex square root simplifications. Remember that practice is key to mastering this concept. Work through several examples, applying the steps outlined above, and you'll soon feel confident in simplifying various square roots. Don't hesitate to use a calculator for approximation, but strive to understand the underlying mathematical principles for a deeper grasp of the subject.