Square Root of 1408

The square root is the inverse of the square of the number. 1408 is not a perfect square. The square root of 1408 is expressed in both radical and exponential form. In radical form, it is expressed as √1408, whereas (1408)^(1/2) in exponential form. √1408 ≈ 37.514, 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 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 1408 is broken down into its prime factors.

Step 1: Finding the prime factors of 1408 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 11: 2^7 x 11

Step 2: Now we found out the prime factors of 1408. The second step is to make pairs of those prime factors. Since 1408 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely.

Therefore, calculating 1408 using prime factorization is not straightforward.

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 1408, we need to group it as 08 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 x 3 = 9, which is less than 14. Now the quotient is 3. After subtracting 9 from 14, the remainder is 5.

Step 3: Now let us bring down 08, which is the new dividend. 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 the value of n such that 6n x n ≤ 508. Let us consider n as 8, now 68 x 8 = 544, which is greater than 508. So we try n as 7, 67 x 7 = 469, which is less than 508.

Step 5: Subtract 469 from 508, and the difference is 39, and the quotient becomes 37.

Step 6: 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 3900.

Step 7: Now we need to find the new divisor that is 745 because 745 x 5 = 3725.

Step 8: Subtracting 3725 from 3900, we get the result of 175.

Step 9: 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 √1408 is approximately 37.51.

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 1408 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1408.

The smallest perfect square less than 1408 is 1369, and the largest perfect square greater than 1408 is 1444. √1408 falls somewhere between 37 and 38.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using this formula, (1408 - 1369) / (1444 - 1369) = 39 / 75 ≈ 0.52. 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 37 + 0.52 = 37.52.

So the square root of 1408 is approximately 37.52.

• 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 a principal square root.

• Prime factorization: Prime factorization is breaking down a number into its basic prime number factors.

• Long division method: A mathematical technique used to find the square root of a non-perfect square by using a series of division steps.