Square Root of 663

The square root is the inverse of the square of a number. 663 is not a perfect square. The square root of 663 is expressed in both radical and exponential forms. In the radical form, it is expressed as √663, whereas in exponential form, it is (663)^(1/2). √663 ≈ 25.751, 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 the 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 663 is broken down into its prime factors.

Step 1: Finding the prime factors of 663 Breaking it down, we get 3 x 13 x 17.

Step 2: Now we found out the prime factors of 663. The next step is to make pairs of those prime factors. Since 663 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 663 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 663, we group them as 63 and 6.

Step 2: Now, we need to find n whose square is ≤ 6. We can say n = 2 because 2^2 = 4 is less than 6. The quotient is 2, and after subtracting, the remainder is 2.

Step 3: Now, let us bring down 63, making the new dividend 263. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

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

Step 5: The next step is finding 4n × n ≤ 263. Let us consider n as 5; now, 45 × 5 = 225.

Step 6: Subtract 225 from 263; the difference is 38, and the quotient is 25.

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 3800.

Step 8: Find the new divisor, which is 509, because 509 × 7 = 3563.

Step 9: Subtracting 3563 from 3800 gives us a remainder of 237.

Step 10: Now, the quotient is 25.7.

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

So the square root of √663 ≈ 25.75.

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 663 using the approximation method.

Step 1: We have to find the closest perfect squares around √663. The smallest perfect square less than 663 is 625, and the largest perfect square greater than 663 is 676. So, √663 falls somewhere between 25 and 26.

Step 2: Now, apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Using the formula: (663 - 625) ÷ (676 - 625) = 38 ÷ 51 ≈ 0.745 Using the formula, we identified the decimal part of our square root. The next step is adding the integer part we got initially to the decimal number, which is 25 + 0.745 ≈ 25.75.

So, the square root of 663 is approximately 25.75.

• 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 25 is a perfect square because it is 5^2.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 18 is 2 × 3 × 3.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.