Square Root of 1682

The square root is the inverse of the square of the number. 1682 is not a perfect square. The square root of 1682 is expressed in both radical and exponential form. In the radical form, it is expressed as √1682, whereas (1682)^(1/2) in the exponential form. √1682 ≈ 41.011, 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 1682 is broken down into its prime factors:

Step 1: Finding the prime factors of 1682 Breaking it down, we get 2 × 3 × 281: 2¹ × 3¹ × 281¹

Step 2: Now we found out the prime factors of 1682. The second step is to make pairs of those prime factors. Since 1682 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1682 using prime factorization is impossible.

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 1682, we need to group it as 16 and 82.

Step 2: Now we need to find n whose square is closest to 16. We can say n as ‘4’ because 4 × 4 = 16. Now the quotient is 4 and after subtracting 16 - 16 the remainder is 0.

Step 3: Now let us bring down 82 which is the new dividend. Add the old divisor with the same number 4 + 4, we get 8 which will be our new divisor.

Step 4: We need to find a number that when multiplied by the new divisor gives a product less than or equal to 82. Let us consider n as 1, now 81 × 1 = 81.

Step 5: Subtract 82 from 81 and the difference is 1, and the quotient is 41.

Step 6: Since the remainder is not zero, 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 100.

Step 7: Now we need to find the new divisor, which will be 82. We need to find a number that when multiplied by 82 gives a product less than or equal to 100. The closest we can use is 1.

Step 8: Subtracting 82 from 100 we get the result 18.

Step 9: The quotient is now 41.0 and we continue with additional decimal places to get a more precise value.

So the square root of √1682 is approximately 41.011.

The approximation method is another method for finding 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 1682 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1682.

The smallest perfect square less than 1682 is 1600 and the largest perfect square greater than 1682 is 1764.

√1682 falls somewhere between 40 and 42.

Step 2: Now we need to apply the formula:

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

Using the formula, (1682 - 1600) ÷ (1764 - 1600) ≈ 0.511

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 which is 41 + 0.011 = 41.011, so the square root of 1682 is approximately 41.011.

• Square root: A square root is the inverse of a square. Example: 4² = 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 has more prominence due to its uses in the real world. This is the reason it is also known as the principal square root.
   
• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 1682 is 2 × 3 × 281.
   
• Approximation: The process of finding a value that is close enough to the right answer, usually within a specified range. For example, the approximate value of √1682 is 41.011.