Square Root of 268

The square root is the inverse operation of squaring a number. 268 is not a perfect square. The square root of 268 can be expressed in both radical and exponential forms. In radical form, it is expressed as √268, whereas in exponential form it is expressed as (268)^(1/2). √268 ≈ 16.3707, which is an irrational number because it cannot be expressed as a simple fraction of two integers.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 268, the long-division method and approximation method are more suitable. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number involves expressing it as a product of its prime factors. Now let us look at how 268 is broken down into its prime factors:

Step 1: Finding the prime factors of 268 Breaking it down, we get 2 x 2 x 67: 2^2 x 67

Step 2: Now we found out the prime factors of 268. Since 268 is not a perfect square, the digits cannot be grouped into pairs. Therefore, calculating √268 using prime factorization directly is not possible.

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

Step 1: To begin with, we group the numbers from right to left. In the case of 268, we group it as 68 and 2.

Step 2: Find n such that n^2 is closest to 2. We choose n = 1 because 1^2 is less than or equal to 2. The quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 68 to get the new dividend 168. Add the old divisor 1 to itself to get the new divisor 2.

Step 4: We now have 2n as the new divisor, and we need to find n such that 2n × n ≤ 168. Let us consider n as 6, which gives us 26 × 6 = 156.

Step 5: Subtract 156 from 168 to get the remainder 12, and the quotient is 16.

Step 6: Since the dividend is less than the divisor, we add a decimal point and two zeroes to the dividend. The new dividend is 1200.

Step 7: Find the new divisor. We get 327 because 327 × 3 = 981.

Step 8: Subtract 981 from 1200, leaving us with a remainder of 219.

Step 9: The quotient is now 16.3.

Step 10: Continue this process until you achieve the desired precision. The square root of √268 is approximately 16.37.

The approximation method is another method for finding square roots, and it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 268 using the approximation method.

Step 1: Identify the closest perfect squares around √268. The smallest perfect square less than 268 is 256, and the largest perfect square greater than 268 is 289. √268 falls between 16 and 17.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (268 - 256) / (289 - 256) = 12 / 33 ≈ 0.36 Adding this to the base value of 16, we get 16 + 0.36 = 16.36, so the square root of 268 is approximately 16.36.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root of 16 is √16 = 4.

• Irrational number: An irrational number cannot be written as a simple fraction of two integers (p/q where q ≠ 0).

• Principal square root: Although a number can have both positive and negative square roots, the positive one is known as the principal square root.

• Prime factorization: Breaking down a number into its prime number factors. For example, the prime factorization of 268 is 2 × 2 × 67.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because 4 × 4 = 16.