Simplify The Square Root Of 90

Simplifying the Square Root of 90: A Comprehensive Guide

Simplifying square roots might seem daunting at first, but with a systematic approach and a solid understanding of prime factorization, it becomes a straightforward process. This comprehensive guide will walk you through simplifying the square root of 90, explaining the underlying concepts and providing practical examples to solidify your understanding. We'll cover various methods, ensuring you can tackle similar problems with confidence.

Understanding Square Roots and Prime Factorization

Before diving into simplifying √90, let's refresh our understanding of fundamental concepts.

What is a Square Root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 (√25) is 5 because 5 x 5 = 25. Not all numbers have perfect square roots (integers). Numbers like √90 are irrational, meaning their decimal representation continues infinitely without repeating.

The Importance of Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors—numbers divisible only by 1 and themselves. This is crucial for simplifying square roots because it allows us to identify perfect squares hidden within the number. Perfect squares are numbers that are the product of a number multiplied by itself (e.g., 4, 9, 16, 25).

Method 1: Prime Factorization to Simplify √90

This is the most common and arguably the most reliable method for simplifying square roots.

1. Find the Prime Factors of 90: We start by finding the prime factorization of 90. We can do this using a factor tree:

       90
      /  \
     2   45
        /  \
       3   15
          /  \
         3    5
   

   Therefore, the prime factorization of 90 is 2 x 3 x 3 x 5, or 2 x 3² x 5.

2. Identify Perfect Squares: Notice that we have a pair of 3s (3²). This is a perfect square (3 x 3 = 9).

3. Simplify the Square Root: Now we can rewrite √90 using the prime factorization:

   √90 = √(2 x 3² x 5)

   Since √(a x b) = √a x √b, we can separate the terms:

   √90 = √2 x √(3²) x √5

   We know that √(3²) = 3, so we simplify:

   √90 = 3√(2 x 5) = 3√10

   Therefore, the simplified form of √90 is 3√10.

Method 2: Using Perfect Square Factors

This method is a shortcut if you can quickly identify perfect square factors of the number.

1. Find Perfect Square Factors: Look for perfect square numbers that divide evenly into 90. We know that 9 (3²) is a factor of 90 (90 / 9 = 10).

2. Rewrite the Square Root: We can rewrite √90 as:

   √90 = √(9 x 10)

3. Separate the Terms: Again, using the property √(a x b) = √a x √b:

   √90 = √9 x √10

4. Simplify: Since √9 = 3, we get:

   √90 = 3√10

   Once again, the simplified form is 3√10.

Why Prime Factorization is Preferred

While the second method is quicker if you spot the perfect square factor easily, prime factorization is a more robust and reliable approach. Larger numbers may have multiple perfect square factors, making the second method more complex. Prime factorization ensures you always find the most simplified form, regardless of the number's size or complexity.

Approximating the Value of √90

While we've simplified √90 to 3√10, it's helpful to understand its approximate decimal value.

Since √10 is between √9 (which is 3) and √16 (which is 4), we know that √10 is approximately 3.16. Therefore:

3√10 ≈ 3 x 3.16 ≈ 9.48

Using a calculator, the actual value of √90 is approximately 9.4868. Our approximation is quite close!

Advanced Applications and Further Exploration

Understanding square root simplification is fundamental to various mathematical fields, including:

• Algebra: Simplifying expressions involving square roots is crucial for solving equations and simplifying formulas.
• Geometry: Calculating lengths, areas, and volumes often involves simplifying square roots. Consider the Pythagorean theorem (a² + b² = c²) – you'll frequently encounter square roots.
• Trigonometry: Trigonometric functions and their inverses often yield results involving square roots.
• Calculus: Derivatives and integrals frequently involve operations on square roots.

Practical Examples: Simplifying Other Square Roots

Let's apply what we've learned to simplify a few more square roots:

Example 1: Simplify √72

1. Prime factorization of 72: 2³ x 3²
2. √72 = √(2³ x 3²) = √(2² x 2 x 3²) = 2 x 3√2 = 6√2

Example 2: Simplify √128

1. Prime factorization of 128: 2⁷
2. √128 = √(2⁷) = √(2⁶ x 2) = 2³√2 = 8√2

Example 3: Simplify √180

1. Prime factorization of 180: 2² x 3² x 5
2. √180 = √(2² x 3² x 5) = 2 x 3√5 = 6√5

Conclusion: Mastering Square Root Simplification

Simplifying square roots, particularly a seemingly complex one like √90, becomes manageable with a methodical approach. By understanding prime factorization and applying the principles outlined in this guide, you'll be able to tackle various square root simplification problems with ease and confidence. Remember, practice is key; the more examples you work through, the more proficient you'll become. This fundamental skill will prove invaluable in your mathematical journey. Keep exploring, keep practicing, and you'll master the art of simplifying square roots!