Square Root of 2160

The square root is the inverse of the square of the number. 2160 is not a perfect square. The square root of 2160 is expressed in both radical and exponential form. In radical form, it is expressed as √2160, whereas (2160)^(1/2) in exponential form. √2160 ≈ 46.4758, 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 2160 is broken down into its prime factors:

Step 1: Finding the prime factors of 2160 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 5 x 3: 2^4 x 3^3 x 5

Step 2: Now we found out the prime factors of 2160. The second step is to make pairs of those prime factors. Since 2160 is not a perfect square, therefore the digits of the number can’t be grouped in pairs without a remainder.

Therefore, calculating √2160 using prime factorization involves leaving an unpaired factor.

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 2160, we need to group it as 21 and 60.

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

Step 3: Now let us bring down 60, making the new dividend 560. Add the old divisor with the same number 4 + 4, we get 8 which will be our new divisor.

Step 4: The next step is finding 8n x n ≤ 560. Let us consider n as 6, now 86 x 6 = 516.

Step 5: Subtract 516 from 560, the difference is 44, and the quotient is 46.

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 4400.

Step 7: Now we need to find the new divisor that is 929 because 929 x 9 = 8361.

Step 8: Subtracting 8361 from 44000 we get the result 559.

Step 9: Now the quotient is 46.4.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √2160 is approximately 46.48.

The approximation method is another method for finding the square roots, and 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 2160 using the approximation method.

Step 1: Now we have to find the closest perfect squares surrounding 2160. The smallest perfect square less than 2160 is 2025, and the largest perfect square greater than 2160 is 2304. √2160 falls somewhere between 45 and 48.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Going by the formula (2160 - 2025) ÷ (2304 - 2025) = 0.48. 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 45 + 0.48 = 45.48.

So the square root of 2160 is approximately 46.48.

• 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, 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: Prime factorization is the process of expressing a number as the product of its prime factors. Example: 2160 = 2^4 x 3^3 x 5.

• Long division method: The long division method is a step-by-step process used to find the square root of numbers, especially when they are not perfect squares.