Square Root of 1537

The square root is the inverse of the square of the number. 1537 is not a perfect square. The square root of 1537 is expressed in both radical and exponential form. In the radical form, it is expressed as √1537, whereas 1537^(1/2) in exponential form. √1537 = 39.197, 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 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 1537 is broken down into its prime factors.

Step 1: Finding the prime factors of 1537 Breaking it down, we find that 1537 is a product of smaller prime numbers: 13 x 118, 13 x 2 x 59.

Step 2: Now we found out the prime factors of 1537. The second step is to make pairs of those prime factors. Since 1537 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1537 using prime factorization is not feasible.

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 1537, we need to group it as 37 and 15.

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 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 37, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we need to find the value of n such that 6n x n is less than or equal to 637.

Step 5: The next step is finding 6n x n ≤ 637. Let us consider n as 9, now 69 x 9 = 621.

Step 6: Subtract 621 from 637, the difference is 16, and the quotient is 39.

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 1600.

Step 8: Now we need to find the new divisor that is 391 because 391 x 4 = 1564.

Step 9: Subtracting 1564 from 1600, we get the result 36.

Step 10: Now the quotient is 39.1

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

So the square root of √1537 is approximately 39.197.

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 1537 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1537.

The smallest perfect square less than 1537 is 1521, and the largest perfect square greater than 1537 is 1600. √1537 falls somewhere between 39 and 40.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1537 - 1521) ÷ (1600 - 1521) = 16 ÷ 79 ≈ 0.2025

Using the formula, we identify the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 39 + 0.2025 ≈ 39.2025. So the square root of 1537 is approximately 39.2.

• 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.

• 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.

• Long division method: A method used to find the square root of a non-perfect square number by dividing the number into pairs of digits, starting from the decimal point.

• Approximation method: A method for estimating the square root of a number by finding nearby perfect squares and interpolating between them.