Square Root of 2156

The square root is the inverse of the square of the number. 2156 is not a perfect square. The square root of 2156 is expressed in both radical and exponential form. In the radical form, it is expressed as √2156, whereas (2156)^(1/2) in the exponential form. √2156 ≈ 46.432, 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 2156 is broken down into its prime factors.

Step 1: Finding the prime factors of 2156.

Breaking it down, we get 2 x 2 x 7 x 7 x 11: 2^2 x 7^2 x 11.

Step 2: Now we found out the prime factors of 2156. The second step is to make pairs of those prime factors. Since 2156 is not a perfect square, calculating 2156 using prime factorization directly 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, we need to group the numbers from right to left. In the case of 2156, we need to group it as 56 and 21.

Step 2: Now we need to find n whose square is less than or equal to 21. We can say n is '4' because 4 x 4 = 16, which is less than 21. Now the quotient is 4, and after subtracting 16 from 21, the remainder is 5.

Step 3: Now let us bring down 56, which is the new dividend. Add the old divisor with the same number 4 + 4 = 8, which will be our new divisor.

Step 4: We now have 8n as the new divisor. We need to find a digit n such that 8n x n ≤ 556. If n = 6, then 86 x 6 = 516.

Step 5: Subtract 516 from 556; the difference is 40.

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 4000.

Step 7: Now we need to find the new divisor, which is 927 because 927 x 4 = 3708.

Step 8: Subtracting 3708 from 4000, we get the result 292.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose there is no decimal value; continue till the remainder is zero.

So the square root of √2156 ≈ 46.43.

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 2156 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √2156. The smallest perfect square less than 2156 is 2025 (45^2), and the largest perfect square greater than 2156 is 2209 (47^2). So, √2156 falls somewhere between 46 and 47.

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

Using the formula: (2156 - 2025) / (2209 - 2025) = 131 / 184 ≈ 0.71. Add this to the base integer 46: 46 + 0.71 = 46.71. So the square root of 2156 is approximately 46.71.

• 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 often more relevant in real-world applications. That is why it is also known as the principal square root.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example: 1, 4, 9, 16, 25 are perfect squares.
   
• Long Division Method: A method used to divide large numbers and find the square roots of non-perfect squares by determining successive approximations.