Square Root of 266

The square root is the inverse of squaring a number. 266 is not a perfect square. The square root of 266 is expressed in both radical and exponential form. In radical form, it is written as √266, whereas in exponential form it is expressed as (266)^(1/2). √266 ≈ 16.30951, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is suitable for perfect square numbers. For non-perfect squares like 266, methods like the long-division method and approximation method are used. Let's explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number is the product of its prime factors. Let's see how 266 is broken down into its prime factors:

Step 1: Find the prime factors of 266. Breaking it down, we get 2 x 7 x 19.

Step 2: Since 266 is not a perfect square, its prime factors cannot be grouped into pairs.

Therefore, calculating √266 using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect squares. In this method, we find the closest perfect square number to the given number. Let's learn how to find the square root using the long division method, step by step:

Step 1: Begin by grouping the digits from right to left. For 266, group as 66 and 2.

Step 2: Find n whose square is ≤ 2. Here, n is 1 because 1 x 1 = 1 ≤ 2. Subtract 1 from 2 to get a remainder of 1.

Step 3: Bring down 66 to make the new dividend 166.

Step 4: Double the divisor (1) to get 2.

Step 5: Find n such that 2n x n ≤ 166. n is 6 because 26 x 6 = 156.

Step 6: Subtract 156 from 166 to get a remainder of 10.

Step 7: Add a decimal point, bring down two zeros to make the new dividend 1000.

Step 8: Find n for 326n x n ≤ 1000. n is 3 because 326 x 3 = 978.

Step 9: Subtract 978 from 1000 to get 22.

Step 10: The quotient is 16.3.

Step 11: Continue until two decimal places are found or until the remainder is zero.

So, √266 ≈ 16.31.

The approximation method is another way to find square roots, providing an easy way to estimate the square root of a number. Let's use this method for 266:

Step 1: Identify the closest perfect squares around 266. The smallest perfect square is 256 (16^2) and the largest is 289 (17^2). √266 falls between 16 and 17.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Applying the formula: (266 - 256) / (289 - 256) = 10 / 33 = 0.303. Adding this to the lower bound, 16 + 0.303 ≈ 16.303.

Thus, the square root of 266 is approximately 16.31.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse operation is finding the square root: √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction of two integers.
   
• Decimal: A decimal is a number that contains a whole number and a fractional part separated by a decimal point, like 16.31.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Perfect square: A number that is the square of an integer. For example, 256 is a perfect square because it is 16^2.