Square Root of 1440

The square root is the inverse of the square of the number. 1440 is not a perfect square. The square root of 1440 is expressed in both radical and exponential form. In the radical form, it is expressed as √1440, whereas (1440)^(1/2) in the exponential form. √1440 ≈ 37.94733, 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 1440 is broken down into its prime factors:

Step 1: Finding the prime factors of 1440 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5: 2^5 × 3^2 × 5

Step 2: Now we found out the prime factors of 1440. The second step is to make pairs of those prime factors. Since 1440 is not a perfect square, therefore the digits of the number can’t be grouped in pairs perfectly.

Therefore, calculating 1440 using prime factorization yields an approximate value.

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 1440, we need to group it as 40 and 14.

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

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

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 540. Let us consider n as 9, now 69 × 9 = 621, which is greater than 540. Trying n as 8, we get 68 × 8 = 544, which is slightly greater, so n is 7 here.

Step 6: Subtract 544 from 540, the difference is -4, and the quotient is 37.

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 4000.

Step 8: Now we need to find the new divisor that is 759 because 759 × 5 = 3795.

Step 9: Subtracting 3795 from 4000, we get the result 205.

Step 10: Now the quotient is 37.9.

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 √1440 is approximately 37.95.

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 1440 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1440.

The smallest perfect square less than 1440 is 1369, and the largest perfect square less than 1440 is 1521. √1440 falls somewhere between 37 and 39.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1440 - 1369) ÷ (1521 - 1369) = 71 / 152 = 0.467.

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 37 + 0.467 = 37.467, so the square root of 1440 is approximately 37.95.

• 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 a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Prime factorization: It is the process of expressing a number as the product of its prime factors.

• Approximation: It is a method used to find an estimated value that is close to the actual value but not exact.