Square Root of 1732

The square root is the inverse of the square of the number. 1732 is not a perfect square. The square root of 1732 is expressed in both radical and exponential form. In the radical form, it is expressed as √1732, whereas (1732)^(1/2) in the exponential form. √1732 ≈ 41.613, 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 1732 is broken down into its prime factors

Step 1: Finding the prime factors of 1732. Breaking it down, we get 2 x 2 x 433: 2^2 x 433^1.

Step 2: Now we found out the prime factors of 1732. The second step is to make pairs of those prime factors. Since 1732 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1732 using prime factorization is not feasible for finding its 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 1732, we need to group it as 32 and 17.

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 lesser than or equal to 17. Now the quotient is 4, and after subtracting 16 from 17, the remainder is 1.

Step 3: Now let us bring down 32, making it 132, which is the new dividend. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: We need to find a digit for n such that 8n x n ≤ 132. If n is 1, then 81 x 1 = 81.

Step 5: Subtract 81 from 132. The difference is 51, and the new quotient is 41.

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 5100.

Step 7: Now we need to find a new digit for n such that 820 x n x n is close to 5100. Continuing this process, we refine the value of the square root.

So the square root of √1732 is approximately 41.613.

The approximation method is another method for finding the 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 1732 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1732.

The smallest perfect square less than 1732 is 1600 (40^2), and the largest perfect square more than 1732 is 1764 (42^2).

√1732 falls somewhere between 41 and 42.

Step 2: Now we need to apply the formula to find an approximate value.

Using the approximation formula:

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

(1732 - 1600) ÷ (1764 - 1600) = 132 ÷ 164 ≈ 0.805.

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 + 0.805 ≈ 41.805. The square root of 1732 is approximately 41.805.

• 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, 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 a principal square root.
   
• Prime factorization: Prime factorization involves breaking down a composite number into a product of its prime factors.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing and refining the quotient step by step.