Square Root of 1666

The square root is the inverse of the square of the number. 1666 is not a perfect square. The square root of 1666 is expressed in both radical and exponential form. In radical form, it is expressed as √1666, whereas (1666)^(1/2) in exponential form. √1666 ≈ 40.812, 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 1666 is broken down into its prime factors:

Step 1: Finding the prime factors of 1666 Breaking it down, we get 2 x 7 x 7 x 17: 2^1 x 7^2 x 17^1

Step 2: Now we found out the prime factors of 1666. The second step is to make pairs of those prime factors. Since 1666 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1666 using prime factorization for an exact square root is not possible.

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 1666, we need to group it as 66 and 16.

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

Step 3: Now let us bring down 66, 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: Now we get 8n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 66. Let us consider n as 8, now 8 x 8 = 64.

Step 6: Subtract 66 from 64, the difference is 2, and the quotient is 48.

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 200.

Step 8: Now we need to find the new divisor that is 81 because 481 x 1 = 481. Continue these steps until we get two numbers after the decimal point.

The square root of √1666 is approximately 40.81.

The approximation method is another method for finding square roots, and 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 1666 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √1666.

The smallest perfect square less than 1666 is 1600 (40^2), and the largest perfect square greater than 1666 is 1681 (41^2).

√1666 falls somewhere between 40 and 41.

Step 2: Now we need to apply the formula:

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

Applying the formula: (1666 - 1600) / (1681 - 1600) = 66 / 81 ≈ 0.81

Using the formula, we identified the decimal point of our square root.

The next step is adding the whole number part to the decimal number, which is 40 + 0.81 = 40.81.

So the square root of 1666 is approximately 40.81.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. 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, but the positive square root is known as the principal square root and is often used in real-world applications.
   
• Prime factorization: Breaking down a number into its basic prime numbers that multiply together to give the original number.
   
• Long division method: A technique used to find the square root of a non-perfect square number through a series of division and estimates.