Square Root of 1540

The square root is the inverse of the square of the number. 1540 is not a perfect square. The square root of 1540 is expressed in both radical and exponential forms. In the radical form, it is expressed as √1540, whereas (1540)^(1/2) in the exponential form. √1540 ≈ 39.24256, 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 1540 is broken down into its prime factors.

Step 1: Finding the prime factors of 1540 Breaking it down, we get 2 × 2 × 5 × 7 × 11: 2^2 × 5^1 × 7^1 × 11^1

Step 2: Now we found out the prime factors of 1540. The second step is to make pairs of those prime factors. Since 1540 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1540 using prime factorization is impossible.

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 1540, we need to group it as 40 and 15.

Step 2: Now we need to find n whose square is 15. We can say n is ‘3’ because 3 × 3 is lesser than or equal to 15. Now the quotient is 3 after subtracting 9 (3 × 3) from 15, the remainder is 6.

Step 3: Now let us bring down 40, which is the new dividend. Add the old divisor with the same number 3 + 3, and we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum with a placeholder, making it 60. Now we get 60n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 60n × n ≤ 640. Let us consider n as 1, now 60 × 1 = 60.

Step 6: Subtract 60 from 640, the difference is 580, and the quotient is 31.

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 58000.

Step 8: Now we need to find the new divisor that is 392 because 392 × 9 = 3528.

Step 9: Subtracting 3528 from 5800, we get the result 2272.

Step 10: Now the quotient is 39.2

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 √1540 is approximately 39.24.

The approximation method is another method for finding the 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 1540 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1540

The smallest perfect square less than 1540 is 1521, and the largest perfect square more than 1540 is 1600.

√1540 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).

Going by the formula (1540 - 1521) ÷ (1600-1521) = 0.24

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 + 0.24 = 39.24, so the square root of 1540 is approximately 39.24.

• 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, 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.
   
• 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.