Square Root of 661

The square root is the inverse of the square of the number. 661 is not a perfect square. The square root of 661 is expressed in both radical and exponential form. In the radical form, it is expressed as √661, whereas (661)^(1/2) in the exponential form. √661 ≈ 25.704, 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, the prime factorization method is not used for non-perfect square numbers where long-division method and approximation method 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 661 is broken down into its prime factors.

Step 1: Finding the prime factors of 661 Breaking it down, we get 661 = 19 x 37. Since 661 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 661 using prime factorization for an exact square root is not feasible.

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 661, we need to group it as 61 and 6.

Step 2: Now we need to find n whose square is 6. We can say n as ‘2’ because 2 x 2 = 4, which is lesser than 6. Now the quotient is 2 and after subtracting 4 from 6, the remainder is 2.

Step 3: Now let us bring down 61, making the new dividend 261. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n ≤ 261. Let us consider n as 6, now 46 x 6 = 276, which is too large. Trying n as 5 gives 45 x 5 = 225, which is suitable.

Step 5: Subtract 225 from 261, the difference is 36, and the current quotient is 25.

Step 6: 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, making it 3600.

Step 7: We need to find the new divisor and corresponding n. Continuing this process allows us to approximate √661 to 25.704.

The approximation method is another method for finding square roots, and 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 661 using the approximation method.

Step 1: Identify the closest perfect squares to 661. The smallest perfect square below 661 is 625 (25^2) and the largest perfect square above 661 is 676 (26^2). Thus, √661 falls between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (661 - 625) ÷ (676 - 625) ≈ 0.704. Adding the value to the integer part gives us 25 + 0.704 = 25.704, so the square root of 661 is approximately 25.704.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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, q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots, however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: The expression of a number as the product of its prime factors. For example, 661 = 19 x 37.
   
• Long division method: A procedure used to find the square root of a number by dividing it into groups of two digits and estimating each digit of the root one by one.