Square Root of 605

The square root is the inverse of the square of the number. 605 is not a perfect square. The square root of 605 is expressed in both radical and exponential forms. In the radical form, it is expressed as √605, whereas in exponential form it is (605)^(1/2). √605 ≈ 24.5972, 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 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 605 is broken down into its prime factors:

Step 1: Finding the prime factors of 605 Breaking it down, we get 5 × 11 × 11: 5^1 × 11^2

Step 2: Now we found out the prime factors of 605. The second step is to make pairs of those prime factors. Since 605 is not a perfect square, all prime factors cannot be perfectly paired. Therefore, calculating 605 using prime factorization will not yield an exact whole number square root.

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 605, we need to group it as 05 and 6.

Step 2: Now we need to find n whose square is closest to 6. We can say n as ‘2’ because 2 × 2 = 4, which is lesser than or equal to 6. The quotient is 2, subtracting 4 from 6 gives the remainder 2.

Step 3: Now let us bring down 05 to make the new dividend 205. Add the old divisor with the same number 2 + 2, we get 4 which will be our new divisor.

Step 4: Now we get 4n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 205. Let us consider n as 5, now 4 × 5 × 5 = 100.

Step 6: Subtract 100 from 205, the difference is 105, and the quotient is 25.

Step 7: Since the dividend is more than the divisor, we continue with decimal places. Add a decimal point and bring down two zeroes to the dividend, making it 10500.

Step 8: Now we need to find the new divisor, 495, because 495 × 2 = 990.

Step 9: Subtracting 990 from 1050 gives a remainder of 1050.

Step 10: The quotient becomes 24.5.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue till the remainder is zero. So the square root of √605 is approximately 24.60.

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 605 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √605. The smallest perfect square less than 605 is 576, and the largest perfect square greater than 605 is 625. √605 falls somewhere between 24 and 25.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (605 - 576) ÷ (625 - 576) = 29 ÷ 49 ≈ 0.59. 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 24 + 0.59 = 24.59, so the square root of 605 is approximately 24.59.

• 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, 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 is used more often due to its significance in the real world. That is why it is also known as the 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.

• Long Division Method: This is a method used to find the square root of a non-perfect square number by performing division through a series of steps to find the square root value.