Square Root of 632

The square root is the inverse of the square of the number. 632 is not a perfect square. The square root of 632 is expressed in both radical and exponential form. In the radical form, it is expressed as √632, whereas (632)^(1/2) in the exponential form. √632 ≈ 25.15, 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 632 is broken down into its prime factors.

Step 1: Finding the prime factors of 632 Breaking it down, we get 2 x 2 x 2 x 79: 2^3 x 79^1

Step 2: Now we found out the prime factors of 632. The second step is to make pairs of those prime factors. Since 632 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 632 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 632, we need to group it as 32 and 6.

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

Step 3: Now let us bring down 32 which is the new dividend. Add the old divisor with the same number 2 + 2 we get 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 x n ≤ 232. Let us consider n as 5, now 45 x 5 = 225.

Step 6: Subtract 225 from 232, the difference is 7, 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 700.

Step 8: Now we need to find the new divisor that is 501 because 501 x 1 = 501.

Step 9: Subtracting 501 from 700 we get the result 199.

Step 10: Now the quotient is 25.1.

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

So the square root of √632 is approximately 25.15.

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 632 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √632. The smallest perfect square less than 632 is 625, and the largest perfect square greater than 632 is 676. √632 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (632 - 625) / (676 - 625) = 7 / 51 ≈ 0.137 Using the formula we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 25 + 0.137 ≈ 25.137, so the square root of 632 is approximately 25.137.

• Square root: A square root is the inverse of a square. For example, 5^2=25, and the inverse of the square is the square root, that is √25 = 5.
   
• 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, the positive square root is more commonly used due to its applications in the real world. This is known as the principal square root.
   
• Prime factorization: Prime factorization involves expressing a number as a product of its prime factors.
   
• Decimal: A decimal is a number that includes a whole number and a fraction represented together, such as 7.86, 8.65, and 9.42.