Square Root of 4671

The square root is the inverse of the square of the number. 4671 is not a perfect square. The square root of 4671 is expressed in both radical and exponential form. In the radical form, it is expressed as √4671, whereas (4671)^(1/2) in the exponential form. √4671 ≈ 68.3498, 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, for non-perfect square numbers, 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 4671 is broken down into its prime factors.

Step 1: Finding the prime factors of 4671

Breaking it down, we get 3 x 3 x 3 x 173: 3^3 x 173^1

Step 2: Now we found out the prime factors of 4671. The second step is to make pairs of those prime factors. Since 4671 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √4671 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 4671, we need to group it as 71 and 46.

Step 2: Now we need to find a number whose square is less or equal to 46. We can choose 6 because 6 x 6 = 36, which is less than 46. Now the quotient is 6, after subtracting 36 from 46, the remainder is 10.

Step 3: Bring down 71, making the new dividend 1071. Double the quotient 6 to get 12, which will be part of our new divisor.

Step 4: The new divisor will be 12n, and we need to find n such that 12n x n ≤ 1071. Let us consider n as 8, now 128 x 8 = 1024.

Step 5: Subtract 1024 from 1071, the difference is 47, and the quotient is 68.

Step 6: 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 4700.

Step 7: Continue the process to find more decimal places. The square root of 4671 can be approximated to √4671 ≈ 68.3498.

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 4671 using the approximation method.

Step 1: Find the closest perfect squares around √4671. The smallest perfect square less than 4671 is 4624 (68^2), and the largest perfect square greater than 4671 is 4761 (69^2). Therefore, √4671 falls between 68 and 69.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square) Using the formula: (4671 - 4624) / (4761 - 4624) = 47 / 137 ≈ 0.343 Adding the decimal to the nearest lower perfect square root: 68 + 0.343 = 68.343

So the square root of 4671 is approximately 68.343.

• 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 usually the positive square root that is used most often in real-world applications. This is known as the principal square root.

• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.

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