Square Root of 1632

The square root is the inverse of the square of the number. 1632 is not a perfect square. The square root of 1632 is expressed in both radical and exponential form. In the radical form, it is expressed as √1632, whereas \(1632^{1/2}\) in the exponential form. √1632 ≈ 40.398, 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 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 1632 is broken down into its prime factors.

Step 1: Finding the prime factors of 1632 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 3 x 17: (2^5 times 3^1 times 17^1)

Step 2: Now we found out the prime factors of 1632. The second step is to make pairs of those prime factors. Since 1632 is not a perfect square, therefore the digits of the number can’t be grouped in pairs perfectly.

Therefore, calculating 1632 using prime factorization for an exact square root 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 1632, we need to group it as 32 and 16.

Step 2: Now we need to find a number whose square is less than or equal to 16. We can say this number is 4 because (4 times 4 = 16). Now the quotient is 4, and after subtracting 16 - 16, the remainder is 0.

Step 3: Now let us bring down 32, which is the new dividend. Add the old divisor with the same number, \(4 + 4 = 8\), 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 times n leq 32). Let us consider n as 3, now (8 times 3 = 24).

Step 6: Subtracting 32 from 24, the difference is 8, and the quotient is 43.

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 800.

Step 8: Now we need to find the new divisor, which is 406 because (406 times 1 = 406).

Step 9: Subtracting 406 from 800, we get the result 394.

Step 10: Now the quotient is 40.39.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue until the remainder is zero.

So the square root of √1632 is approximately 40.398.

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 1632 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1632.

The smallest perfect square less than 1632 is 1600, and the largest perfect square greater than 1632 is 1681.

√1632 falls somewhere between 40 and 41.

Step 2: Now we need to apply the formula: \((\text{Given number} - \text{smallest perfect square}) / (\text{Greater perfect square} - \text{smallest perfect square})\).

Going by the formula: \((1632 - 1600) \div (1681 - 1600) = 0.398\).

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 40 + 0.398 = 40.398, so the square root of 1632 is approximately 40.398.

• Square root: A square root is the inverse of the square. For 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 the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 1632 is \(2^5 \times 3 \times 17\).
   
• Long division method: A method used to find the square root of a non-perfect square by dividing the number into groups and using a step-by-step division process to approximate the square root.