Square Root of 1810

The square root is the inverse of the square of a number. 1810 is not a perfect square. The square root of 1810 is expressed in both radical and exponential form. In radical form, it is expressed as √1810, whereas in exponential form, it is expressed as (1810)^(1/2). √1810 ≈ 42.53875, 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 typically used for perfect square numbers. However, for non-perfect square numbers, 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 1810 is broken down into its prime factors.

Step 1: Finding the prime factors of 1810 Breaking it down, we get 2 x 5 x 181: 2^1 x 5^1 x 181^1

Step 2: Now that we found the prime factors of 1810, the next step is to make pairs of those prime factors. Since 1810 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 1810 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 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, group the numbers from right to left. In the case of 1810, group it as 18 and 10.

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

Step 3: Bring down 10, making the new dividend 210. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. 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 x n ≤ 210. Let us consider n as 2; now 82 x 2 = 164.

Step 6: Subtract 164 from 210, the difference is 46, and the quotient is 42.

Step 7: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 4600.

Step 8: Find the new divisor which is 849 because 849 x 5 = 4245.

Step 9: Subtracting 4245 from 4600 gives the result 355.

Step 10: Now the quotient is 42.5.

Step 11: Continue these steps until you have two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √1810 is approximately 42.54.

The approximation method is another method for finding square roots, and it is an easy way to find the square root of a given number. Now let us learn how to find the square root of 1810 using the approximation method.

Step 1: Find the closest perfect squares to √1810. The smallest perfect square less than 1810 is 1764, and the largest perfect square greater than 1810 is 1849. √1810 falls somewhere between 42 and 43.

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

Using this formula, (1810 - 1764) / (1849 - 1764) = 46 / 85 = 0.54. 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: 42 + 0.54 = 42.54.

So the square root of 1810 is approximately 42.54.

• Square root: A square root is the inverse of squaring a number. 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 is primarily used due to its applications in the real world. This is known as the principal square root.

• Prime factorization: It is the process of expressing a number as a product of its prime factors.

• Approximation method: A method used to estimate the value of a square root by identifying the closest perfect squares and finding a value between them.