Square Root of 1805

The square root is the inverse of the square of a number. 1805 is not a perfect square. The square root of 1805 is expressed in both radical and exponential form. In the radical form, it is expressed as √1805, whereas (1805)^(1/2) in the exponential form. √1805 ≈ 42.5047, 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 employed. 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 1805 is broken down into its prime factors.

Step 1: Finding the prime factors of 1805. Breaking it down, we get 5 x 19 x 19: 5 x 19^2.

Step 2: Now, we found out the prime factors of 1805. The second step is to make pairs of those prime factors. Since 1805 is not a perfect square, calculating its square root using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers 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 1805, we need to group it as 05 and 18.

Step 2: Now, we need to find n whose square is less than or equal to 18. We can say n as ‘4’ because 4 x 4 = 16, which is less than 18. Now the quotient is 4, after subtracting 16 from 18 the remainder is 2.

Step 3: Now let us bring down 05 to make it 205, 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: The new divisor will be 8n. We need to find the value of n such that 8n x n ≤ 205. Let us consider n as 2, now 82 x 2 = 164.

Step 5: Subtract 164 from 205, the difference is 41, and the quotient is 42. 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 4100.

Step 7: Now, we need to find the new divisor that is 849 because 849 x 4 = 3396.

Step 8: Subtracting 3396 from 4100 we get the result 704.

Step 9: The quotient is now 42.4.

Step 10: Continue doing these steps until we get a sufficient number of decimal places.

So the square root of √1805 ≈ 42.5047.

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 approximate the square root of 1805.

Step 1: Now we have to find the closest perfect squares of √1805. The smallest perfect square less than 1805 is 1764 (42^2) and the largest perfect square greater than 1805 is 1849 (43^2). √1805 falls somewhere between 42 and 43.

Step 2: Now, apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

By the formula (1805 - 1764) ÷ (1849 - 1764) ≈ 0.5047.

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 42 + 0.5047 ≈ 42.5047.

So, the square root of 1805 is approximately 42.5047.

• 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, the positive square root is typically used in practical applications. That is why it is known as the principal square root.

• Prime factorization: The expression of a number as the product of its prime factors.

• Long division method: A step-by-step approach for finding the square root of a number by division, especially useful for non-perfect squares.