What Is Square Root Of 45

What is the Square Root of 45? A Deep Dive into Square Roots and Simplification

The question, "What is the square root of 45?" might seem simple at first glance. However, exploring this seemingly straightforward mathematical concept opens a door to a deeper understanding of square roots, prime factorization, and simplifying radical expressions. This article will not only answer the question directly but also provide a comprehensive guide to understanding and working with square roots, particularly those involving non-perfect squares like 45.

Understanding Square Roots

Before diving into the specifics of the square root of 45, let's establish a fundamental understanding of what a square root is. The square root of a number is a value that, when multiplied by itself (squared), gives the original number. For example:

• The square root of 9 (√9) is 3, because 3 * 3 = 9.
• The square root of 16 (√16) is 4, because 4 * 4 = 16.

These are examples of perfect squares, numbers that have whole number square roots. However, many numbers don't have whole number square roots. 45 is one such number. This leads us to the concept of simplifying radicals.

Prime Factorization: The Key to Simplifying √45

To simplify the square root of 45, we utilize prime factorization. This involves breaking down a number into its prime factors – numbers that are only divisible by 1 and themselves. Let's factorize 45:

45 = 9 * 5

Since 9 is not a prime number, we further factorize it:

9 = 3 * 3

Therefore, the prime factorization of 45 is 3 * 3 * 5, or 3² * 5.

Simplifying the Square Root of 45

Now that we have the prime factorization of 45 (3² * 5), we can simplify the square root:

√45 = √(3² * 5)

Because the square root of a product is the product of the square roots, we can rewrite this as:

√45 = √3² * √5

Since √3² = 3, the simplified form of √45 is:

√45 = 3√5

This means that 3√5, when multiplied by itself, equals 45. We have expressed the square root of 45 in its simplest radical form. It's important to note that this is not a decimal approximation; it's the exact value.

Decimal Approximation of √45

While 3√5 is the exact simplified form, we can also find a decimal approximation using a calculator:

√45 ≈ 6.708

This is an approximation, as the square root of 45 is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation goes on forever without repeating.

Working with Square Roots: Further Exploration

Understanding the square root of 45 provides a solid foundation for working with other square roots, particularly those involving simplification. Let's explore some related concepts and examples:

More Examples of Simplifying Radicals

Let's try simplifying a few more square roots using prime factorization:

• √72: 72 = 2³ * 3² = 2² * 2 * 3² => √72 = √(2² * 3² * 2) = 2 * 3 * √2 = 6√2

• √128: 128 = 2⁷ = 2⁶ * 2 = (2³)² * 2 => √128 = √((2³)² * 2) = 2³√2 = 8√2

• √180: 180 = 2² * 3² * 5 => √180 = √(2² * 3² * 5) = 2 * 3 * √5 = 6√5

These examples demonstrate the consistent application of prime factorization and the rules of square roots to simplify radical expressions.

Adding and Subtracting Radicals

Once you have simplified radicals, you can add and subtract them, but only if the radicand (the number inside the square root) is the same:

• 2√5 + 3√5 = 5√5
• 7√2 - 4√2 = 3√2

However, you cannot simplify 2√5 + 3√2 because the radicands are different.

Multiplying and Dividing Radicals

Multiplying and dividing radicals involves multiplying or dividing the radicands and simplifying the result:

• √2 * √3 = √6
• √15 / √5 = √3

Rationalizing the Denominator

If you have a square root in the denominator of a fraction, you need to rationalize the denominator. This involves multiplying both the numerator and the denominator by the square root in the denominator:

• 1/√2: Multiply by √2/√2 to get √2/2.

Applications of Square Roots

Square roots appear in various areas of mathematics and beyond, including:

• Geometry: Calculating the hypotenuse of a right-angled triangle using the Pythagorean theorem (a² + b² = c²).
• Physics: Solving problems related to velocity, acceleration, and distance.
• Engineering: Designing structures and calculating forces.
• Computer graphics: Creating realistic images and animations.

Understanding square roots is essential for anyone working with these fields.

Conclusion: Mastering Square Roots

This comprehensive guide has explored the square root of 45 in detail, showing how to simplify it to its simplest radical form (3√5) and providing a decimal approximation (approximately 6.708). More importantly, it's highlighted the broader principles of square roots, prime factorization, and simplifying radical expressions. Mastering these techniques is crucial for success in various mathematical and scientific endeavors. Remember to always break down the number into its prime factors before attempting to simplify a square root. This methodical approach will ensure accuracy and efficiency in your calculations. By understanding the underlying principles, you can confidently tackle more complex square root problems and appreciate the elegance and practicality of this fundamental mathematical concept.