Square Root of 226

The square root is the inverse operation of squaring a number. Since 226 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is expressed as √226, whereas in exponential form it is expressed as (226)^(1/2). √226 ≈ 15.0333, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

For perfect squares, the prime factorization method is typically used. However, for non-perfect squares like 226, methods such as long division and approximation are employed. Let us delve into these methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization method involves expressing a number as a product of its prime factors. Let's break down 226 into its prime factors:

Step 1: Finding the prime factors of 226 Breaking it down, we get 2 x 113: 2^1 x 113^1

Step 2: Since 226 is not a perfect square, the digits cannot be paired.

Therefore, calculating √226 using prime factorization is not feasible.

The long division method is well-suited for non-perfect squares. This method involves finding the closest perfect square for the given number. Let's find the square root using this method, step by step:

Step 1: Group the digits in pairs from right to left. For 226, consider it as 26 and 2.

Step 2: Find the largest number whose square is less than or equal to 2. Here, n = 1. Subtract 1^2 from 2, leaving a remainder of 1.

Step 3: Bring down the next pair, 26, making it 126. Double the divisor (1), giving 2, to form the new divisor.

Step 4: Find n such that 2n x n ≤ 126. Using n = 5, we have 25 x 5 = 125.

Step 5: Subtract 125 from 126, leaving a remainder of 1. The quotient is 15.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros, making it 100.

Step 7: The new divisor becomes 300 (from 2 x 15), and n is 3 since 303 x 3 = 909.

Step 8: Subtract 909 from 1000, leaving 91.

Step 9: Continue this process to achieve two decimal places.

Thus, √226 ≈ 15.03.

The approximation method provides an easy way to find the square root of a number. Here's how to find the square root of 226 using this method:

Step 1: Identify the closest perfect squares to √226. The nearest perfect square less than 226 is 225, and the nearest greater is 256. So, √226 is between 15 and 16.

Step 2: Apply the formula: (Given number - smaller perfect square)/(Greater perfect square - smaller perfect square). Using the formula (226 - 225) ÷ (256 - 225) = 1/31 ≈ 0.0323. Add this to the smaller perfect square root: 15 + 0.0323 ≈ 15.0323, so √226 ≈ 15.03.

• Square root: The square root is the inverse operation of squaring a number. For example: 4² = 16, and the square root of 16 is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. For instance, √226 is irrational.
   
• Principal square root: The positive square root of a number is known as the principal square root.
   
• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 226 is 2 x 113.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing them systematically.