Square Root of 261

The square root is the inverse of the square of the number. 261 is not a perfect square. The square root of 261 is expressed in both radical and exponential form. In the radical form, it is expressed as √261, whereas (261)^(1/2) in the exponential form. √261 ≈ 16.155, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

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

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 261 is broken down into its prime factors.

Step 1: Finding the prime factors of 261 Breaking it down, we get 3 x 3 x 29: 3^2 x 29

Step 2: Now we found out the prime factors of 261. The second step is to make pairs of those prime factors. Since 261 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 261 using prime factorization is impossible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check 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 need to group the numbers from right to left. In the case of 261, we group it as 61 and 2.

Step 2: Now we need to find n whose square is 2. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 61, making it the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 161. Let us consider n as 8, now 28 x 8 = 224

Step 6: Subtract 161 from 224, the difference is 63, and the quotient is 16

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 6300.

Step 8: Now we need to find the new divisor. Let us consider n as 2, 322 x 2 = 644

Step 9: Subtracting 644 from 6300, we get the result 5656.

Step 10: Now the quotient is 16.1

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue until the remainder is zero.

So the square root of √261 is approximately 16.15.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 261 using the approximation method.

Step 1: Now we have to find the closest perfect square of √261. The smallest perfect square less than 261 is 256, and the largest perfect square greater than 261 is 289. √261 falls somewhere between 16 and 17.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (261 - 256) / (289 - 256) = 5 / 33 ≈ 0.1515 Adding the calculated decimal to the whole number 16, we get 16 + 0.1515 = 16.1515, so the square root of 261 is approximately 16.15.

• Square root: A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 16 is a perfect square because it is 4².
   
• Non-perfect square: A non-perfect square is a number that cannot be expressed as the square of an integer. Example: 261 is a non-perfect square.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups of two digits from right to left and following a step-by-step division process.