Square Root of 4025

The square root is the inverse of the square of the number. 4025 is not a perfect square. The square root of 4025 is expressed in both radical and exponential form. In the radical form, it is expressed as √4025, whereas (4025)^(1/2) in the exponential form. √4025 = 63.466, 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 4025 is broken down into its prime factors:

Step 1: Finding the prime factors of 4025 Breaking it down, we get 5 x 5 x 13 x 31: 5^2 x 13 x 31

Step 2: Now we have found the prime factors of 4025. The second step is to make pairs of those prime factors. Since 4025 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 4025 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 4025, we need to group it as 25 and 40.

Step 2: Now we need to find n whose square is 40. We can say n as ‘6’ because 6 x 6 = 36, which is lesser than or equal to 40. Now the quotient is 6, after subtracting 40 - 36, the remainder is 4.

Step 3: Now let us bring down 25, which makes the new dividend 425. Add the old divisor (6) with itself to get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 425. Let us consider n as 3, now 123 x 3 = 369.

Step 6: Subtract 425 from 369; the difference is 56, and the quotient is 63.

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 5600.

Step 8: Now we need to find the new divisor that is 634 because 6344 x 4 = 25376.

Step 9: Subtracting 25376 from 56000, we get the result 30624.

Step 10: Now the quotient is 63.4.

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 √4025 is approximately 63.47.

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 4025 using the approximation method.

Step 1: Find the closest perfect square of √4025. The smallest perfect square less than 4025 is 3969, and the largest perfect square greater than 4025 is 4096. √4025 falls somewhere between 63 and 64.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (4025 - 3969) ÷ (4096 - 3969) = 56 / 127 = 0.44. 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 63 + 0.44 = 63.44, so the square root of 4025 is approximately 63.44.

• 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, 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.
   
• Prime factorization: The process of determining the prime numbers that multiply together to give a particular integer.
   
• Decimal number: A number with a whole number and a fractional part, separated by a decimal point. For example: 7.86, 8.65, and 9.42 are decimals.