Square Root of 4625

The square root is the inverse of the square of the number. 4625 is not a perfect square. The square root of 4625 is expressed in both radical and exponential form. In the radical form, it is expressed as √4625, whereas in exponential form, it is (4625)^(1/2). √4625 ≈ 67.997, 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 4625 is broken down into its prime factors.

Step 1: Finding the prime factors of 4625

Breaking it down, we get 5 x 5 x 5 x 37: 5^3 x 37

Step 2: Now that we have found the prime factors of 4625, we attempt to make pairs of those prime factors. Since 4625 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √4625 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 4625, we can group it as 46 and 25.

Step 2: Now we need to find n whose square is closest to 46. We can say n as ‘6’ because 6 x 6 = 36 is less than 46. Now the quotient is 6, and after subtracting 36 from 46, the remainder is 10.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number 6 + 6 to get 12, which will be our new divisor.

Step 4: The new divisor will be 12n. We need to find the value of n such that 12n x n is less than or equal to 1025.

Step 5: The next step is finding 12n x n ≤ 1025. Let us consider n as 8, now 128 x 8 = 1024.

Step 6: Subtract 1024 from 1025; the difference is 1, and the quotient is 68.

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 100.

Step 8: Now we need to find the new divisor that is 1369 because 1369 x 7 = 9583.

Step 9: Subtracting 9583 from 10000 gives us the result 417.

Step 10: Now the quotient is 67.9.

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

So the square root of √4625 is approximately 67.997.

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 4625 using the approximation method.

Step 1: Now we have to find the closest perfect square of √4625. The smallest perfect square less than 4625 is 4624, and the largest perfect square greater than 4625 is 4761. √4625 falls somewhere between 68 and 69.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (4625 - 4624) / (4761 - 4624) = 1/137 = 0.0073. 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 68 - 0.0073 ≈ 67.993, so the square root of 4625 is approximately 67.993.

• 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.

• 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.

• Prime factorization: Prime factorization involves expressing a number as the product of its prime factors. For example, the prime factorization of 4625 is 5^3 x 37.