What's The Square Root Of 128

What's the Square Root of 128? A Deep Dive into Square Roots and Simplification

The question, "What's the square root of 128?" seems simple enough. However, understanding the answer fully involves delving into the fascinating world of square roots, prime factorization, and simplifying radical expressions. This comprehensive guide will not only provide the answer but also equip you with the knowledge to tackle similar problems with confidence.

Understanding Square Roots

Before we tackle the square root of 128, let's establish a foundational understanding of what a square root actually is. The square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 9 is 3 because 3 x 3 = 9. We represent the square root using the radical symbol (√).

Key Concepts:

• Perfect Squares: These are numbers that result from squaring an integer (whole number). Examples include 1 (1²), 4 (2²), 9 (3²), 16 (4²), and so on.
• Irrational Numbers: Many square roots are irrational numbers – they cannot be expressed as a simple fraction and their decimal representation goes on forever without repeating.
• Rational Numbers: Some square roots are rational numbers, meaning they can be expressed as a fraction of two integers. This occurs when the number inside the square root is a perfect square.

Finding the Square Root of 128: A Step-by-Step Approach

1. Prime Factorization: The first step in simplifying the square root of 128 is to find its prime factorization. This means breaking down the number into its prime number components (numbers only divisible by 1 and themselves).

   128 = 2 x 64 = 2 x 2 x 32 = 2 x 2 x 2 x 16 = 2 x 2 x 2 x 2 x 8 = 2 x 2 x 2 x 2 x 2 x 4 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2⁷

2. Pairing Factors: Now that we have the prime factorization (2⁷), we look for pairs of identical factors. Each pair of identical factors can be brought out of the square root as a single factor.

   √128 = √(2⁷) = √(2⁶ x 2)

3. Simplifying the Expression: We can now simplify the expression:

   √(2⁶ x 2) = √(2⁶) x √2 = 2³ x √2 = 8√2

Therefore, the simplified form of the square root of 128 is 8√2.

This means that 8√2 multiplied by itself (8√2 x 8√2) equals 128. Let's verify:

(8√2) x (8√2) = 64 x 2 = 128

Decimal Approximation

While 8√2 is the exact simplified form, you might need a decimal approximation for practical applications. Using a calculator, we find that:

√128 ≈ 11.3137

Therefore, the square root of 128 is approximately 11.3137.

Further Exploration: Working with Square Roots

Let's explore additional scenarios involving square roots to solidify your understanding:

Simplifying Other Square Roots

Let's try another example: Find the square root of 72.

1. Prime Factorization: 72 = 2 x 36 = 2 x 2 x 18 = 2 x 2 x 2 x 9 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

2. Pairing Factors: We have one pair of 2s and one pair of 3s.

3. Simplification: √72 = √(2³ x 3²) = √(2² x 2 x 3²) = √2² x √3² x √2 = 2 x 3 x √2 = 6√2

Therefore, the simplified square root of 72 is 6√2.

Adding and Subtracting Square Roots

You can add and subtract square roots only if they have the same radicand (the number inside the square root). For example:

3√5 + 2√5 = 5√5

However, you cannot directly add 3√5 and 2√2.

Multiplying and Dividing Square Roots

Multiplying and dividing square roots involves multiplying or dividing the radicands:

√a x √b = √(a x b) √a / √b = √(a/b)

For example:

√2 x √8 = √16 = 4 √18 / √2 = √9 = 3

Applications of Square Roots

Square roots have numerous applications in various fields, including:

• Geometry: Calculating the diagonal of a square or rectangle, the length of a hypotenuse using the Pythagorean theorem (a² + b² = c²).
• Physics: Calculating velocity, acceleration, and other physical quantities.
• Engineering: Solving equations and designing structures.
• Computer Science: Used in algorithms and data structures.
• Finance: Calculating returns on investments.

Conclusion: Mastering Square Roots

Understanding square roots, including how to simplify them through prime factorization, is a fundamental skill in mathematics and many related disciplines. The ability to simplify expressions like the square root of 128 (8√2) demonstrates a grasp of core mathematical concepts and opens doors to tackling more complex problems. By mastering the techniques outlined in this guide, you'll be well-equipped to confidently handle square roots in any context. Remember to practice regularly, working through various examples to solidify your understanding and build your problem-solving skills. This will not only improve your mathematical proficiency but also enhance your ability to approach and solve problems in a logical and structured manner.