Square Root of 6125

The square root is the inverse of the square of the number. 6125 is not a perfect square. The square root of 6125 is expressed in both radical and exponential form. In the radical form, it is expressed as √6125, whereas (6125)^(1/2) in the exponential form. √6125 ≈ 78.262379, 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 6125 is broken down into its prime factors:

Step 1: Finding the prime factors of 6125 Breaking it down, we get 5 x 5 x 5 x 7 x 7: 5^3 x 7^2

Step 2: Now we found out the prime factors of 6125. The second step is to make pairs of those prime factors. Since 6125 is not a perfect square, the digits of the number can’t be grouped in complete pairs.

Therefore, calculating √6125 using prime factorization gives an approximate result.

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 6125, we need to group it as 61 and 25.

Step 2: Now we need to find n whose square is less than or equal to 61. We can say n as '7' because 7 x 7 = 49, which is less than 61. Now the quotient is 7, and after subtracting, the remainder is 12.

Step 3: Bring down 25, making the new dividend 1225. Add the old divisor with the same number, 7 + 7 = 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, and we need to find the value of n.

Step 5: The next step is finding 14n x n ≤ 1225. Let us consider n as 8, now 148 x 8 = 1184.

Step 6: Subtract 1225 from 1184; the difference is 41, 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 4100.

Step 8: Now we need to find the new divisor that is 785 because 785 x 5 = 3925.

Step 9: Subtracting 3925 from 4100, we get the result 175.

Step 10: Now the quotient is 78.2.

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

So the square root of √6125 ≈ 78.26.

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 6125 using the approximation method.

Step 1: Now we have to find the closest perfect square of √6125.

The smallest perfect square less than 6125 is 6084 and the largest perfect square greater than 6125 is 6241. √6125 falls somewhere between 78 and 79.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (6125 - 6084) / (6241 - 6084) = 41 / 157 = 0.261

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 78 + 0.261 = 78.261.

So the square root of 6125 is approximately 78.261.

• 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. This is why it is also known as the principal square root.

• Prime factorization: Finding the prime numbers that multiply together to get the original number. For example, the prime factorization of 6125 is 5 x 5 x 5 x 7 x 7.

• 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.