Square Root of 1404

The square root is the inverse of the square of the number. 1404 is not a perfect square. The square root of 1404 is expressed in both radical and exponential form. In radical form, it is expressed as √1404, whereas in exponential form it is (1404)^(1/2). √1404 ≈ 37.478, 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, for non-perfect square numbers, the long division method and approximation method are often used. Let us now learn these 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 1404 is broken down into its prime factors:

Step 1: Finding the prime factors of 1404 Breaking it down, we get 2 x 2 x 3 x 3 x 3 x 13.

Step 2: Now that we have found the prime factors of 1404, the next step is to make pairs of these prime factors. Since 1404 is not a perfect square, the digits of the number cannot be grouped into pairs completely.

Therefore, calculating √1404 using prime factorization requires approximation.

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 1404, we need to group it as 14 and 04.

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

Step 3: Bring down 04, making the new dividend 504. 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 6n. We need to find n such that 6n × n ≤ 504. Let us consider n as 8. Now, 68 x 8 = 544.

Step 5: Since 544 is greater than 504, we try n as 7. So, 67 x 7 = 469, which is less than 504.

Step 6: Subtract 469 from 504. The difference is 35, and the quotient is 37.

Step 7: Since the remaining dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 3500.

Step 8: Find the new divisor: 746 x 4 = 2984.

Step 9: Subtracting 2984 from 3500 gives 516.

Step 10: The quotient is now 37.4.

Step 11: Continue doing these steps until we achieve the desired level of precision.

Thus, the square root of √1404 is approximately 37.48.

The approximation method is another approach for finding square roots, and it is an easy method to estimate the square root of a given number. Let us learn how to find the square root of 1404 using the approximation method.

Step 1: We have to find the closest perfect squares of √1404.

The smallest perfect square less than 1404 is 1369 (37^2) and the largest perfect square greater than 1404 is 1444 (38^2). Therefore, √1404 falls between 37 and 38.

Step 2: Now we need to apply the formula: (Given number - Smaller perfect square) / (Larger perfect square - Smaller perfect square).

Using this formula, (1404 - 1369) / (1444 - 1369) = 35 / 75 ≈ 0.467. Adding this value to the smaller base value we found gives 37 + 0.467 ≈ 37.467.

Hence, the square root of 1404 is approximately 37.467.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 x 4 = 16.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction p/q, where p and q are integers and q ≠ 0. Examples include √2 and π.

• Principal square root: The principal square root of a non-negative number is its non-negative square root. For example, the principal square root of 9 is 3.

• Prime factorization: This is the expression of a number as the product of its prime numbers. For example, the prime factorization of 28 is 2 x 2 x 7.

• Approximation method: This method is used to estimate values, such as square roots, by finding the nearest known values and interpolating between them.