Square Root of 4160

The square root is the inverse of the square of the number. 4160 is not a perfect square. The square root of 4160 is expressed in both radical and exponential forms. In the radical form, it is expressed as √4160, whereas (4160)^(1/2) in the exponential form. √4160 ≈ 64.495, 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 4160 is broken down into its prime factors.

Step 1: Finding the prime factors of 4160 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 13 x 8: 2^4 x 5 x 13 x 8

Step 2: Now we found out the prime factors of 4160. The second step is to make pairs of those prime factors. Since 4160 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating √4160 using prime factorization is complex, and we use other methods for approximation.

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 4160, we need to group it as 60 and 41.

Step 2: Now we need to find n whose square is closest to 41. We can say n as ‘6’ because 6 x 6 is 36, which is lesser than 41. Now the quotient is 6 and after subtracting 36 from 41, the remainder is 5.

Step 3: Now let us bring down 60, which is the new dividend. Add the old divisor with the same number 6 + 6, we get 12, which will be our new divisor.

Step 4: The new divisor is 12n, and we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 560. Let us consider n as 4, now 12 x 4 = 48, so 48 x 4 = 192.

Step 6: Subtract 192 from 560, the difference is 368, and the quotient is 64.

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 zeros to the dividend. Now the new dividend is 36800.

Step 8: Now we need to find the new divisor. If we assume the next digit in the quotient is 9, we calculate 1289 × 9 = 11601, which is less than 36800.

Step 9: Subtract 11601 from 36800, and we get the result 25199.

Step 10: Now the quotient is 64.9

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 √4160 is approximately 64.49

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 4160 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √4160. The smallest perfect square of 4160 is 4096, and the largest perfect square of 4160 is 4225. √4160 falls somewhere between 64 and 65.

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 (4160 - 4096) ÷ (4225 - 4096) = 0.5128 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 64 + 0.5128 = 64.5128, so the square root of 4160 is approximately 64.51

• 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.
   
• Long division method: A method used to find the square root of non-perfect squares, where numbers are divided into groups and systematically divided to approach the square root.
   
• Approximation method: A method used to estimate the square root by finding nearby perfect squares and using a formula to calculate a decimal approximation.