Square Root of 2837

The square root is the inverse of the square of a number. 2837 is not a perfect square. The square root of 2837 is expressed in both radical and exponential form. In the radical form, it is expressed as √2837, whereas (2837)^(1/2) in the exponential form. √2837 ≈ 53.267, 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 and approximation methods are applied. 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 2837 is broken down into its prime factors.

Step 1: Finding the prime factors of 2837. Breaking it down, we get 2837 = 7 x 7 x 41.

Step 2: Now we found the prime factors of 2837. The second step is to make pairs of those prime factors. Since 2837 is not a perfect square, the digits cannot be grouped into pairs perfectly.

Therefore, calculating 2837 using prime factorization alone is not feasible.

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 2837, we need to group it as 37 and 28.

Step 2: Now we need to find n whose square is less than or equal to 28. We can say n is ‘5’ because 5 x 5 = 25, which is less than 28. The quotient is 5 and we subtract 25 from 28, leaving the remainder 3.

Step 3: Bring down 37 to the remainder, making the new dividend 337. Add the old divisor with the result of the previous quotient multiplied by 2, 5 + 5 = 10, which will be the new divisor.

Step 4: Find n such that 10n x n is less than or equal to 337. Let n be 3, so 103 x 3 = 309.

Step 5: Subtract 309 from 337, leaving a remainder of 28. The quotient is now 53.

Step 6: Since the dividend is less than the divisor, add a decimal point to the quotient and two zeros to the dividend, making it 2800.

Step 7: Find the new divisor. Here, 106x is close to 2800, so assume x is 2, where 1062 x 2 = 2124.

Step 8: Subtract 2124 from 2800, leaving the remainder 676.

Step 9: The quotient is now 53.2. Continue the steps until you achieve the desired decimal precision.

So the square root of √2837 is approximately 53.267.

The approximation method is another approach for finding square roots. It's an easy method to find the square root of a given number. Let's learn how to find the square root of 2837 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √2837. The smaller perfect square of 2837 is 2809, and the larger perfect square is 2916. Thus, √2837 falls between 53 and 54.

Step 2: Now apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (2837 - 2809) / (2916 - 2809) ≈ 0.267. Step 3: Add this decimal to the smaller integer square root: 53 + 0.267 = 53.267.

Therefore, the square root of 2837 is approximately 53.267.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16, and the inverse is √16 = 4.
   
• Irrational number: An irrational number cannot be written as a fraction of two integers (p/q), where q ≠ 0.
   
• Principal square root: A number has both positive and negative square roots; the positive one is called the principal square root.
   
• Perfect square: A perfect square is a number that is the square of an integer, like 16, which is 4².
   
• Long division method: A technique used to find the square root of non-perfect squares through repeated division and approximation steps.