Square Root of 1517

The square root is the inverse of the square of a number. 1517 is not a perfect square. The square root of 1517 is expressed in both radical and exponential form. In radical form, it is expressed as √1517, whereas (1517)^(1/2) is the exponential form. √1517 ≈ 38.9504, 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 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 1517 is broken down into its prime factors.

Step 1: Finding the prime factors of 1517 Breaking it down, we get 1517 = 37 x 41.

Step 2: Now that we have found the prime factors of 1517, we realize that since 1517 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 1517 using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect square numbers. In this method, we check for the closest perfect square number to 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 1517, we group it as 15 and 17.

Step 2: Now, we need to find n whose square is less than or equal to 15. We can say n is ‘3’ because 3 x 3 = 9, which is less than 15. Now, the quotient is 3, and after subtracting 9 from 15, the remainder is 6.

Step 3: Now, let us bring down 17, making the new dividend 617. Add the old divisor with itself, 3 + 3, which gives us 6 as the new divisor.

Step 4: The new divisor will be 6n. We need to find the value of n such that 6n x n ≤ 617. Let us consider n as 9, then 69 x 9 = 621.

Step 5: Subtract 621 from 617, the difference is -4, but since this is incorrect, we adjust n to 8, so 68 x 8 = 544.

Step 6: Subtract 544 from 617, the difference is 73, and the quotient is 38.

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 zeroes to the dividend. Now the new dividend is 7300.

Step 8: Now, we need to find the new divisor, which is 389, because 389 x 9 = 3501.

Step 9: Subtracting 3501 from 7300 gives the result 3799.

Step 10: Now, the quotient is 38.9.

Step 11: Continue these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √1517 is approximately 38.95.

The approximation method is another method for finding square roots, and 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 1517 using the approximation method.

Step 1: We have to find the closest perfect squares to √1517.

The smallest perfect square less than 1517 is 1521, and the largest perfect square close to 1517 is 1600. √1517 falls somewhere between 38 and 40.

Step 2: Now, we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1517 - 1521) / (1600 - 1521) = -4 / 79 ≈ -0.0506.

Using the formula, we identified the adjustment to the nearest lower perfect square root. The next step is adjusting the value we got initially to the decimal number: 39 - 0.05 ≈ 38.95, so the square root of 1517 is approximately 38.95.

• 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 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 method is used to find the square root of non-perfect squares by systematically dividing the number and finding its square root to several decimal places.