Square Root of 662

The square root is the inverse of the square of the number. 662 is not a perfect square. The square root of 662 is expressed in both radical and exponential form. In the radical form, it is expressed as √662, whereas (662)^(1/2) in the exponential form. √662 ≈ 25.72936, 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 and approximation methods 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 662 is broken down into its prime factors:

Step 1: Finding the prime factors of 662 Breaking it down, we get 2 × 331: 2^1 × 331^1

Step 2: Now we found out the prime factors of 662. Since 662 is not a perfect square, calculating its square root using prime factorization is not straightforward.

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 662, we group it as 62 and 6.

Step 2: Now we need to find n whose square is closest to 6. We can say n as ‘2’ because 2 × 2 is 4, which is less than or equal to 6. Now the quotient is 2, and after subtracting, the remainder is 2.

Step 3: Bring down 62, making the new dividend 262. Add the old divisor (2) to itself, giving us 4, which will be our new divisor.

Step 4: The next step is finding 4n × n ≤ 262. Let us consider n as 6, now 4 × 6 × 6 = 144.

Step 5: Subtract 144 from 262; the difference is 118.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point, allowing us to add two zeroes to the dividend. Now the new dividend is 11800.

Step 7: Find the new divisor, which is 45, because 457 × 7 = 3199.

Step 8: Subtracting 3199 from 11800, we get 8601.

Step 9: Now the quotient is 25.7.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So the square root of √662 is approximately 25.73.

Approximation method is another method for finding the 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 662 using the approximation method.

Step 1: Now we have to find the closest perfect square of √662. The smallest perfect square less than 662 is 625, and the largest perfect square greater than 662 is 676. √662 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square). Using the formula (662 - 625) ÷ (676 - 625) = 37/51 ≈ 0.72549.

Step 3: Add the initial value to the decimal number: 25 + 0.73 = 25.73, so the square root of 662 is approximately 25.73.

• 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, which 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.
   
• 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.
   
• Approximation method: A method of finding the square root by identifying the closest perfect squares and estimating the value between them.
   
• Long division method: A systematic procedure of dividing a number to find its square root, especially useful for non-perfect squares.