Square Root of 1700

The square root is the inverse of the square of the number. 1700 is not a perfect square. The square root of 1700 is expressed in both radical and exponential form. In the radical form, it is expressed as √1700, whereas (1700)^(1/2) in the exponential form. √1700 ≈ 41.2311, 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 1700 is broken down into its prime factors.

Step 1: Finding the prime factors of 1700 Breaking it down, we get 2 x 2 x 5 x 5 x 17: 2² x 5² x 17

Step 2: Now we found out the prime factors of 1700. The second step is to make pairs of those prime factors. Since 1700 is not a perfect square, therefore the digits of the number can’t be grouped in perfect pairs.

Therefore, calculating 1700 using prime factorization is not straightforward for finding the square root.

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 1700, we need to group it as 17 and 00.

Step 2: Now we need to find n whose square is less than or equal to 17. We can say n as ‘4’ because 4 x 4 = 16, which is less than 17. Now the quotient is 4, and after subtracting 16 from 17, the remainder is 1.

Step 3: Now let us bring down 00, which makes the new dividend 100. 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 the sum of the previous divisor and quotient. Now we get 8n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 100. Let us consider n as 1, now 81 x 1 = 81.

Step 6: Subtract 81 from 100; the difference is 19, and the quotient is 41.

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 1900.

Step 8: Now we need to find the new divisor. Try using 412 because 412 x 4 = 1648.

Step 9: Subtracting 1648 from 1900, we get the result 252.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √1700 is approximately 41.23.

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 1700 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1700.

The smallest perfect square less than 1700 is 1600, and the largest perfect square greater than 1700 is 1764.

√1700 falls somewhere between 40 and 42.

Step 2: Now we need to apply the formula that is

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

Using the formula (1700 - 1600) ÷ (1764 - 1600) = 100 ÷ 164 ≈ 0.61.

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

The next step is adding the value we got initially to the decimal number which is 40 + 0.61 = 40.61, so the square root of 1700 is approximately 40.61.

• 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. That is the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization is the process of breaking down a number into its prime factors.
   
• Approximation method: The approximation method is a technique used to find an approximate value of a square root of a non-perfect square number.