Square Root of 1664

The square root is the inverse of the square of the number. 1664 is not a perfect square. The square root of 1664 is expressed in both radical and exponential form. In the radical form, it is expressed as √1664, whereas in exponential form it is (1664)^(1/2). √1664 ≈ 40.789, 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 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 1664 is broken down into its prime factors.

Step 1: Find the prime factors of 1664 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 13: 2^7 x 13

Step 2: Now we found the prime factors of 1664. The next step is to pair the prime factors. Since 1664 is not a perfect square, the digits of the number can’t be grouped into pairs completely.

Therefore, calculating √1664 using prime factorization alone 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 1664, we need to group it as 64 and 16.

Step 2: Now we need to find n whose square is less than or equal to 16. We can say n is ‘4’ because 4 x 4 = 16. Now the quotient is 4 after subtracting 16-16; the remainder is 0.

Step 3: Now let us bring down 64, which is the new dividend. Add the old divisor with the same number, 4 + 4, to get 8, which will be our new divisor.

Step 4: The new divisor will be 8n, and we need to find the value of n such that 8n x n ≤ 64. Let's consider n as 0.

Step 5: Since the new dividend is 64, we subtract the product of 80 x 0 from 64, giving a remainder of 64.

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 6400.

Step 7: We now need to find the new divisor. We continue the process to find n such that the next product is less than or equal to 6400.

Step 8: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √1664 is approximately 40.789.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 1664 using the approximation method.

Step 1: Find the closest perfect squares to √1664.

The closest perfect squares to 1664 are 1600 and 1681.

√1664 falls somewhere between 40 and 41.

Step 2: Apply the formula

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (1664 - 1600) ÷ (1681 - 1600) ≈ 0.789

Adding this decimal to 40 gives us 40 + 0.789 = 40.789.

Thus, the square root of 1664 is approximately 40.789.

• 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, i.e., √16 = 4.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Approximation: A method for finding an estimated value close to the exact result, often used for non-perfect squares.
   
• Long division method: A step-by-step process used to find the square root of non-perfect squares.
   
• Perfect square: A number that can be expressed as the square of an integer. For example, 64 is a perfect square because it equals 8 x 8.