Square Root of 60

The square root is the inverse of the square of a number. 60 is not a perfect square. The square root of 60 is expressed in both radical and exponential form. In the radical form, it is expressed as √60, whereas (60)^(1/2) in the exponential form. √60 ≈ 7.74597, 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, methods like 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 60 is broken down into its prime factors.

Step 1: Finding the prime factors of 60

Breaking it down, we get 2 x 2 x 3 x 5: 2^2 x 3^1 x 5^1

Step 2: Now we found out the prime factors of 60. The second step is to make pairs of those prime factors. Since 60 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √60 using prime factorization gives us 2√15, which is an irrational number.

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 60, we need to group it as 60.

Step 2: Now we need to find n whose square is less than or equal to 60. We can say n as ‘7’ because 7 x 7 = 49, which is less than 60. Now the quotient is 7, and the remainder is 60 - 49 = 11.

Step 3: Now let us bring down 00, which makes the new dividend 1100.

Step 4: Double the divisor (7) and write it as 14.

Step 5: We need to find a digit x such that 14x x x ≤ 1100. Trying x = 7 gives us 1447 x 7 = 10129, which is too large. Trying x = 6 gives us 146 x 6 = 876, which fits.

Step 6: Subtract 876 from 1100, which gives 224.

Step 7: Since the remainder is not zero, continue the process with the new dividend 22400 and new divisor 1460.

Step 8: The next x is found similarly. After a few more steps, we calculate the square root of 60 as approximately 7.74.

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 60 using the approximation method.

Step 1: Now we have to find the closest perfect squares of 60. The smallest perfect square less than 60 is 49 (7^2), and the largest perfect square greater than 60 is 64 (8^2). √60 falls somewhere between 7 and 8.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square). Going by the formula (60 - 49) / (64 - 49) = 11/15 ≈ 0.733

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 7 + 0.733 = 7.733, so the square root of 60 is approximately 7.73.

• Square root: A square root is the inverse of a square. Example: 4² = 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.
   
• Approximation: The process of finding a value that is close to the actual value, often used for non-perfect squares.
   
• Prime factorization: Breaking down a number into its basic prime number components.
   
• Long division method: A systematic approach for finding the square root of non-perfect squares by using division.