Square Root of 617

The square root is the inverse of the square of the number. 617 is not a perfect square. The square root of 617 is expressed in both radical and exponential form. In the radical form, it is expressed as √617, whereas (617)^(1/2) in the exponential form. √617 ≈ 24.81935, 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 617 is broken down into its prime factors.

Step 1: Finding the prime factors of 617 617 is a prime number. Therefore, it cannot be broken down further into other prime factors.

Step 2: Since 617 is a prime number, it cannot be expressed as a product of smaller prime numbers. The prime factorization method cannot be used to simplify the square root of 617.

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 617, we need to group it as 17 and 6.

Step 2: Now we need to find n whose square is 6. We can say n as '2' because 2 x 2 = 4 is lesser than or equal to 6. Now the quotient is 2; after subtracting 6 - 4, the remainder is 2.

Step 3: Now let us bring down 17, which is the new dividend. Add the old divisor with the same number 2 + 2 we get 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. 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 ≤ 217. Let us consider n as 5, now 45 x 5 = 225, which is greater than 217. So, let's try n as 4. Now 44 x 4 = 176.

Step 6: Subtract 217 from 176; the difference is 41, and the quotient is 24.

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 4100.

Step 8: Now we need to find the new divisor that is 249 because 249 x 9 = 2241.

Step 9: Subtracting 2241 from 4100 we get the result 1859.

Step 10: Now the quotient is approximately 24.8.

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 √617 is approximately 24.819.

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 617 using the approximation method.

Step 1: Now we have to find the closest perfect square of √617. The smallest perfect square less than 617 is 576 and the largest perfect square greater than 617 is 625. √617 falls somewhere between 24 and 25.

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 (617 - 576) / (625 - 576) = 41 / 49 ≈ 0.8367. 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 + 0.8367 ≈ 24.8367, so the square root of 617 is approximately 24.8367.

• 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.

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

• 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.

• Approximation method: A method used to estimate the value of a non-perfect square root by determining which two perfect squares it lies between and then finding a closer value through simple calculations.