Square Root Of 68 In Radical Form

Understanding the Square Root of 68 in Radical Form

The square root of 68, denoted as √68, represents a number that, when multiplied by itself, equals 68. While a calculator will give you a decimal approximation (approximately 8.246), expressing it in its simplest radical form is crucial for many mathematical applications and demonstrates a deeper understanding of number theory. This article will delve into the process of simplifying √68, exploring the underlying concepts, and providing examples to solidify your understanding.

What is a Radical Form?

Before diving into the simplification of √68, let's define what a radical form is. In mathematics, a radical is a symbol (√) used to denote the root of a number. The number inside the radical symbol is called the radicand. A radical is considered to be in its simplest form when the radicand contains no perfect square factors other than 1. This means we need to find the largest perfect square that divides evenly into 68.

Prime Factorization: The Key to Simplification

The most effective method for simplifying radicals involves prime factorization. Prime factorization is the process of breaking down a number into its prime factors—numbers divisible only by 1 and themselves. Let's apply this to 68:

• 68 is an even number, so it's divisible by 2: 68 = 2 x 34
• 34 is also even: 34 = 2 x 17
• 17 is a prime number: Therefore, the prime factorization of 68 is 2 x 2 x 17, or 2² x 17

Simplifying √68

Now that we have the prime factorization of 68 (2² x 17), we can simplify the square root:

√68 = √(2² x 17)

Because √(a x b) = √a x √b, we can rewrite the expression:

√68 = √2² x √17

Since √2² = 2, the simplified radical form is:

√68 = 2√17

This is the simplest radical form because 17 is a prime number and contains no perfect square factors other than 1. We've successfully extracted the perfect square (2²) from the radicand, leaving the remaining prime factor (17) inside the radical.

Understanding Perfect Squares

Understanding perfect squares is fundamental to simplifying radicals. A perfect square is the result of squaring an integer (multiplying an integer by itself). Examples include:

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• and so on...

When simplifying radicals, we look for perfect squares that are factors of the radicand. If we find one, we can extract its square root, simplifying the radical expression.

More Examples of Radical Simplification

Let's practice simplifying more radicals to solidify our understanding:

Example 1: Simplifying √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. Simplification: √72 = √(2³ x 3²) = √(2² x 2 x 3²) = √2² x √3² x √2 = 2 x 3 x √2 = 6√2

Example 2: Simplifying √125

1. Prime Factorization: 125 = 5 x 25 = 5 x 5 x 5 = 5³

2. Simplification: √125 = √(5³) = √(5² x 5) = √5² x √5 = 5√5

Example 3: Simplifying √288

1. Prime Factorization: 288 = 2 x 144 = 2 x 12 x 12 = 2 x (2 x 2 x 3)² = 2 x 2⁴ x 3² = 2⁵ x 3²

2. Simplification: √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 identify and extract perfect square factors from the radicand, resulting in the simplest radical form.

Adding and Subtracting Radicals

Once we understand how to simplify radicals, we can perform operations such as addition and subtraction. However, we can only add or subtract radicals that have the same radicand.

Example:

Simplify 2√17 + 3√17

Since both radicals have the same radicand (17), we simply add the coefficients:

2√17 + 3√17 = 5√17

However, we cannot directly add or subtract radicals with different radicands, such as 2√17 + 3√2. These must remain as separate terms.

Multiplying and Dividing Radicals

Multiplying and dividing radicals involves manipulating the radicands.

Multiplication:

√a x √b = √(a x b)

Example:

√2 x √6 = √(2 x 6) = √12 = √(4 x 3) = 2√3

Division:

√a / √b = √(a / b)

Example:

√18 / √2 = √(18 / 2) = √9 = 3

Applications of Radical Simplification

The simplification of radicals, while seemingly a theoretical exercise, has practical applications across various mathematical fields:

• Geometry: Calculating lengths of sides or diagonals in geometrical shapes often involves simplifying radicals.
• Algebra: Solving quadratic equations sometimes yields solutions in radical form, requiring simplification.
• Calculus: Derivatives and integrals often involve radical expressions that need simplification for easier manipulation.
• Physics: Many physical formulas involve square roots and require simplification for practical calculations.

Conclusion: Mastering Radical Simplification

Simplifying radicals is a fundamental skill in mathematics. By understanding prime factorization and identifying perfect square factors, you can efficiently reduce radical expressions to their simplest form. This process not only enhances mathematical understanding but is also essential for solving complex problems across various disciplines. Remember to always look for the largest perfect square factor to achieve the most simplified radical form. Through consistent practice and application of the techniques outlined in this article, you can master the art of simplifying radical expressions. Remember to break down complex problems into manageable steps using prime factorization, and you'll be well on your way to confidently simplifying any radical expression.