Square Root of 1417

The square root is the inverse of the square of a number. 1417 is not a perfect square. The square root of 1417 can be expressed in both radical and exponential forms. In radical form, it is expressed as √1417, whereas in exponential form, it is written as (1417)^(1/2). √1417 ≈ 37.645, 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 suitable for non-perfect square numbers where long-division and approximation methods 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 1417 is broken down into its prime factors:

Step 1: Finding the prime factors of 1417 Breaking it down, we find it is already a product of prime numbers: 1417 is 1417 x 1, as 1417 is a prime number.

Step 2: Since 1417 is a prime number and not a perfect square, calculating 1417 using prime factorization is not feasible for finding its square root.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers 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 1417, we can group it as 17 and 14.

Step 2: Now we need to find n whose square is less than or equal to 14. We can say n is 3 because 3 x 3 = 9, which is less than 14. Now the quotient is 3, and after subtracting 9 from 14, the remainder is 5.

Step 3: Now let us bring down 17, making the new dividend 517. Add the old divisor (3) with itself to get 6, which will be part of our new divisor.

Step 4: The new divisor will be 6n, where n is a digit such that 6n x n is less than or equal to 517. Let us try n = 8, then 68 x 8 = 544, which is too large. Try n = 7, then 67 x 7 = 469, which is less than 517.

Step 5: Subtract 469 from 517, and the remainder is 48. The quotient is now 37.

Step 6: Since the remainder is not zero, add a decimal point to the quotient, and bring down two zeros to make the new dividend 4800.

Step 7: The new divisor is 674 (67 doubled plus 4), and we find a new digit n such that 674n x n is less than or equal to 4800. Let n be 7, then 6747 x 7 = 4729.

Step 8: Subtract 4729 from 4800, the remainder is 71, and the quotient is 37.7.

Step 9: Continue doing these steps until we get two digits after the decimal point.

The square root of √1417 is approximately 37.65.

The approximation method is another approach 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 1417 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1417.

The smallest perfect square less than 1417 is 1369 (37^2), and the largest perfect square greater than 1417 is 1444 (38^2). √1417 falls somewhere between 37 and 38.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (1417 - 1369) / (1444 - 1369) = 48 / 75 ≈ 0.64. Adding this value to 37 gives us 37 + 0.64 = 37.64, so the square root of 1417 is approximately 37.64.

• Square root: A square root is the inverse of a square. For example, 4² = 16 and the inverse of the square is the square root, which is √16 = 4.

• Prime number: A number greater than 1 that has no positive divisors other than 1 and itself. Example: 2, 3, 5, 7, etc.

• 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.

• Long division method: A technique used to find square roots of non-perfect squares involving a series of steps of division and subtraction.

• Approximation: A method for finding a close value to the actual quantity, often used when an exact value is difficult to obtain.