Square Root of 1679

The square root is the inverse of the square of the number. 1679 is not a perfect square. The square root of 1679 is expressed in both radical and exponential form. In radical form, it is expressed as √1679, whereas in exponential form it is expressed as (1679)^(1/2). √1679 ≈ 40.976, 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 1679 is broken down into its prime factors.

Step 1: Finding the prime factors of 1679 Breaking it down, we get 1679 = 23 x 73.

Step 2: Since 1679 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating √1679 using prime factorization is not feasible for an exact integer result.

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 1679, we need to group it as 79 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, and after subtracting 16 - 16, the remainder is 0.

Step 3: Now let us bring down 79, 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. We need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 79. Let us consider n as 9, now 89 x 9 = 801.

Step 6: Since 801 is greater than 79, we consider n as 8. Now 88 x 8 = 704.

Step 7: Subtract 704 from 790, the difference is 86, and the quotient is 40.8.

Step 8: Add a decimal point, which allows us to add two zeroes to the dividend. Now the new dividend is 8600.

Step 9: Now we need to find the new divisor, which is 409 because 409 x 9 = 3681.

Step 10: Subtracting 3681 from 8600 gives a remainder of 4919.

Step 11: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √1679 is approximately 40.98.

Approximation is another method for finding square roots; it is an easy way to find the square root of a given number. Let us learn how to find the square root of 1679 using the approximation method.

Step 1: Find the closest perfect squares to √1679.

The smallest perfect square less than 1679 is 1600, and the largest perfect square greater than 1679 is 1681.

√1679 falls somewhere between 40 and 41.

Step 2: Now apply the formula:

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

Using the formula (1679 - 1600) ÷ (1681 - 1600) = 0.9875.

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

The next step is adding the value we got initially to the decimal number: 40 + 0.9875 ≈ 40.988, so the square root of 1679 is approximately 40.988.

• 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, which 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, the positive square root is commonly used and is known as the principal square root.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs of digits and using iterative steps.
   
• Prime factorization: The process of breaking down a number into its prime factors, which can be used to simplify square roots for perfect squares.