Square Root of 6025

The square root is the inverse of the square of the number. 6025 is not a perfect square. The square root of 6025 is expressed in both radical and exponential form. In the radical form, it is expressed as √6025, whereas (6025)^(1/2) in the exponential form. √6025 ≈ 77.6182, 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 6025 is broken down into its prime factors:

Step 1: Finding the prime factors of 6025 Breaking it down, we get 5 × 5 × 241: 5^2 × 241^1

Step 2: Now we found out the prime factors of 6025. The second step is to make pairs of those prime factors. Since 6025 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 6025 using prime factorization is not feasible to find its square root directly.

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 6025, we need to group it as 60 and 25.

Step 2: Now we need to find n whose square is 60. We can say n as ‘7’ because 7 × 7 = 49 is lesser than or equal to 60. Now the quotient is 7, and after subtracting 49 from 60, the remainder is 11.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number 7 + 7, we get 14, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 14n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 14n × n ≤ 1125. Let us consider n as 8, now 14 × 8 = 112, and 112 × 8 = 896.

Step 6: Subtract 896 from 1125, the difference is 229, and the quotient is 78.

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 22900.

Step 8: Now we need to find the new divisor, which is 156 (since 2 × 78 = 156) to find n, because 156 × 7 = 1092 and 1092 × 7 = 7644.

Step 9: Subtracting 7644 from 22900, we get the result 15256.

Step 10: Now the quotient is 77.6.

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 √6025 is approximately 77.62.

The 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 6025 using the approximation method.

Step 1: Now we have to find the closest perfect square of √6025.

The smallest perfect square less than 6025 is 5929 (77^2), and the largest perfect square more than 6025 is 6084 (78^2). √6025 falls somewhere between 77 and 78.

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 (6025 - 5929) / (6084 - 5929) = 96 / 155 = 0.619.

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 77 + 0.619 = 77.619, so the square root of 6025 is approximately 77.62.

• 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.

• Approximation method: A mathematical technique used to estimate the square root of a non-perfect square by finding the closest perfect squares.