Simplify The Square Root Of 175

Simplifying the Square Root of 175: A Comprehensive Guide

Simplifying square roots is a fundamental concept in mathematics, crucial for various applications from algebra to calculus. This in-depth guide will walk you through the process of simplifying the square root of 175, explaining the underlying principles and providing you with a solid understanding of square root simplification. We'll go beyond just the solution, exploring the theoretical basis and showing you how to tackle similar problems.

Understanding Square Roots and Prime Factorization

Before diving into simplifying √175, let's revisit the basics. A square root of a number 'x' is a value that, when multiplied by itself, equals 'x'. For instance, the square root of 9 (√9) is 3 because 3 * 3 = 9. However, not all square roots are whole numbers. This is where simplification becomes important.

The key to simplifying square roots lies in prime factorization. Prime factorization is the process of breaking down a number into its prime factors – numbers divisible only by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.). Every number can be expressed as a unique product of prime numbers.

Prime Factorization of 175

Let's find the prime factorization of 175:

1. Start with the smallest prime number, 2. 175 is not divisible by 2 (it's odd).
2. Try the next prime number, 3. 175 is not divisible by 3 (the sum of its digits, 1+7+5=13, is not divisible by 3).
3. Try the next prime number, 5. 175 is divisible by 5 (175 / 5 = 35).
4. Now we have 5 * 35. 35 is also divisible by 5 (35 / 5 = 7).
5. Finally, we have 5 * 5 * 7. 7 is a prime number.

Therefore, the prime factorization of 175 is 5 x 5 x 7, or 5² x 7.

Simplifying √175 Using Prime Factorization

Now that we have the prime factorization of 175 (5² x 7), we can simplify its square root:

√175 = √(5² x 7)

Remember, the square root of a product is the product of the square roots: √(a x b) = √a x √b. Applying this rule:

√(5² x 7) = √5² x √7

Since √5² = 5 (because 5 * 5 = 5²), we get:

√175 = 5√7

Therefore, the simplified form of √175 is 5√7.

Understanding the Logic: Why This Works

The simplification process leverages the properties of square roots and exponents. When a number is squared (raised to the power of 2), and then we take its square root, we effectively cancel out the square and the square root, leaving the base number. This is why √5² = 5.

The number 7, being a prime number and not appearing as a square in the factorization, remains inside the square root. This is because we cannot find a whole number that, when multiplied by itself, equals 7.

Practical Applications and Examples

Simplifying square roots isn't just an abstract mathematical exercise. It's frequently used in:

• Algebra: Solving quadratic equations often involves simplifying square roots to obtain exact solutions.
• Geometry: Calculating the length of diagonals in shapes or the distance between points often requires simplifying square roots derived from the Pythagorean theorem.
• Calculus: Derivatives and integrals frequently involve manipulating square roots and simplifying them for efficient calculations.
• Physics: Many physics formulas, particularly those involving vectors and distances, necessitate simplifying square roots.

Let's look at a few more examples to solidify our understanding:

Example 1: Simplify √72

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

Example 2: Simplify √200

1. Prime factorization of 200: 2 x 2 x 2 x 5 x 5 = 2³ x 5²
2. Simplify the square root: √200 = √(2³ x 5²) = √(2² x 2 x 5²) = √2² x √5² x √2 = 2 x 5 x √2 = 10√2

Example 3: Simplify √288

1. Prime factorization of 288: 2 x 2 x 2 x 2 x 2 x 3 x 3 = 2⁵ x 3²
2. Simplify the square root: √288 = √(2⁵ x 3²) = √(2⁴ x 2 x 3²) = √2⁴ x √3² x √2 = 2² x 3 x √2 = 12√2

These examples demonstrate the consistent application of prime factorization to simplify square roots.

Advanced Techniques and Considerations

While prime factorization is the fundamental method, understanding additional concepts can enhance your ability to simplify square roots:

• Perfect Squares: Recognizing perfect squares (numbers that are the squares of integers, like 4, 9, 16, 25, etc.) within the number you're simplifying can speed up the process. You can directly extract their square roots.

• Largest Perfect Square Factor: Instead of finding the complete prime factorization, you can try to identify the largest perfect square that is a factor of the number. This reduces the number of steps. For instance, in simplifying √175, you might recognize that 25 (5²) is a factor of 175, leading directly to 5√7.

• Rationalizing the Denominator: When dealing with fractions containing square roots in the denominator, you'll need to rationalize the denominator by multiplying both the numerator and denominator by the square root in the denominator. This eliminates the square root from the denominator, resulting in a simplified expression.

Conclusion: Mastering Square Root Simplification

Simplifying square roots is a crucial skill in mathematics. This guide has shown you, step-by-step, how to simplify the square root of 175 and other similar expressions by using prime factorization. By understanding the underlying principles and practicing with various examples, you'll develop confidence and proficiency in simplifying square roots, which will be invaluable in tackling more complex mathematical problems. Remember, the key is mastering prime factorization and applying the properties of square roots consistently. With practice, simplifying square roots will become second nature.