Square Root of 1529

The square root is the inverse of the square of the number. 1529 is not a perfect square. The square root of 1529 is expressed in both radical and exponential form. In the radical form, it is expressed as √1529, whereas (1529)^(1/2) in the exponential form. √1529 ≈ 39.099, 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 1529 is broken down into its prime factors.

Step 1: Finding the prime factors of 1529. Breaking it down, we get 1529 = 1 x 1529 (since 1529 is a prime number).

Step 2: Since 1529 is not a perfect square and it only has two factors, itself and 1, calculating 1529 using prime factorization is not applicable for finding the 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 1529, we can group it as 15 and 29.

Step 2: Now we need to find n whose square is less than or equal to 15. We can say n as ‘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: Bring down 29 to the right of the remainder, making it 629. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be 6n. We need to find n such that 6n x n ≤ 629.

Step 5: Let us consider n as 9; then 69 x 9 = 621.

Step 6: Subtract 621 from 629; the difference is 8, 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 800.

Step 8: The new divisor is 78, because 78 x 1 = 78, which is less than 800.

Step 9: Subtracting 78 from 800 gives us a remainder of 722.

Step 10: Now the quotient is 39.1

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 √1529 is approximately 39.099.

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 1529 using the approximation method.

Step 1: We have to find the closest perfect squares to √1529.

The smallest perfect square less than 1529 is 1521 (39^2) and the largest perfect square greater than 1529 is 1600 (40^2). √1529 falls somewhere between 39 and 40.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1529 - 1521) / (1600 - 1521) ≈ 0.101

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 39 + 0.101 ≈ 39.101.

• 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.

• Prime number: A prime number is a number that has exactly two distinct positive divisors: 1 and itself. Example: 1529 is a prime number.

• Perfect square: A perfect square is an integer that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

• Decimal approximation: A decimal approximation is a numerical estimate of a number, which is not exact but close enough for practical purposes. Example: √1529 ≈ 39.099.