Square Root of 1476

The square root is the inverse of the square of the number. 1476 is not a perfect square. The square root of 1476 is expressed in both radical and exponential form. In the radical form, it is expressed as √1476, whereas (1476)^(1/2) in the exponential form. √1476 ≈ 38.4192, 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 1476 is broken down into its prime factors:

Step 1: Finding the prime factors of 1476 Breaking it down, we get 2 x 2 x 3 x 3 x 41: 2^2 x 3^2 x 41

Step 2: Now we have found the prime factors of 1476. The second step is to make pairs of those prime factors. Since 1476 is not a perfect square, grouping the digits into pairs is not possible.

Therefore, calculating 1476 using prime factorization directly is not feasible.

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 1476, we need to group it as 76 and 14.

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: Now let us bring down 76, which is the new dividend. Add the old divisor with the same number: 3 + 3 = 6, which will be our new divisor.

Step 4: Using the new divisor 6, find the largest single-digit n such that 6n x n is less than or equal to 576. Let n = 9: 69 x 9 = 621, which is too large. Try n = 8: 68 x 8 = 544.

Step 5: Subtract 544 from 576, the difference is 32, and the quotient is 38.

Step 6: Adding a decimal point allows us to add two zeroes to the dividend. Now the new dividend is 3200.

Step 7: Find the new divisor by adding 8 to the previous divisor, giving 76. Determine n such that 760n x n is less than or equal to 3200. The appropriate n is 4, where 764 x 4 = 3056.

Step 8: Subtract 3056 from 3200, resulting in 144.

Step 9: The quotient is now 38.4. Continue this process until the desired decimal precision is reached.

So the square root of √1476 is approximately 38.42.

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 1476 using the approximation method.

Step 1: Identify the closest perfect squares of √1476.

The closest perfect squares are 1444 (38^2) and 1521 (39^2). √1476 falls between 38 and 39.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (1476 - 1444) ÷ (1521 - 1444) = 32 / 77 ≈ 0.416 Adding this decimal to the whole number we started with (38), we get approximately 38.416, so the square root of 1476 is approximately 38.42.

• 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 is more commonly used due to its relevance in the real world. This is known as the principal square root.

• Factors: Factors are numbers that divide a given number completely without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

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