Square Root of 3825

The square root is the inverse of the square of the number. 3825 is not a perfect square. The square root of 3825 is expressed in both radical and exponential form. In the radical form, it is expressed as √3825, whereas (3825)^(1/2) in the exponential form. √3825 ≈ 61.857, 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 3825 is broken down into its prime factors.

Step 1: Finding the prime factors of 3825 Breaking it down, we get 3 × 3 × 5 × 5 × 17: 3^2 × 5^2 × 17

Step 2: Now that we have found the prime factors of 3825, the second step is to make pairs of those prime factors. Since 3825 is not a perfect square, and all factors can't be paired, calculating √3825 using prime factorization directly is not possible.

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 3825, we need to group it as 25 and 38.

Step 2: Now we need to find n whose square is less than or equal to 38. We can take n as ‘6’ because 6 × 6 = 36, which is less than 38. Now the quotient is 6 and after subtracting 36 from 38, the remainder is 2.

Step 3: Now let us bring down 25, which makes the new dividend 225. Add the old divisor with the same number 6 + 6 to get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 225. Let us consider n as 1, then 12 × 1 = 12. Try n = 8, then 12 × 8 = 96 and 121 × 8 = 968.

Step 6: Subtract 968 from 2250 (after adding decimal and two zeroes) to get the difference.

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, making the new dividend 22500.

Step 8: Now we need to find the new divisor. Suppose it is 618 because 618 × 8 = 4944.

Step 9: Subtracting 4944 from 22500, we get the result as 17556.

Step 10: Now the quotient is around 61.8.

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

So the square root of √3825 is approximately 61.86.

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 3825 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3825. The smallest perfect square less than 3825 is 3721, and the largest perfect square greater than 3825 is 3844. √3825 falls somewhere between 61 and 62.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (3825 - 3721) / (3844 - 3721) = 0.857 The next step is adding the value we got initially to the decimal number which is 61 + 0.857 = 61.857, so the square root of 3825 is approximately 61.857.

• Square root: A square root is the inverse of a 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: It is the process of writing a number as the product of its prime factors. For example, the prime factorization of 50 is 2 × 5^2.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.