Square Root of 1216

The square root is the inverse of the square of the number. 1216 is not a perfect square. The square root of 1216 is expressed in both radical and exponential form. In the radical form, it is expressed as √1216, whereas (1216)^(1/2) in the exponential form. √1216 ≈ 34.865, 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, 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 1216 is broken down into its prime factors:

Step 1: Finding the prime factors of 1216 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 19: 2^5 x 19

Step 2: Now we found out the prime factors of 1216. The second step is to make pairs of those prime factors. Since 1216 is not a perfect square, calculating the square root using prime factorization directly 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 1216, we need to group it as 16 and 12.

Step 2: Now we need to find n whose square is less than or equal to 12. We can say n as ‘3’ because 3 x 3 = 9, and 9 is less than 12. Now the quotient is 3 after subtracting 12 - 9, the remainder is 3.

Step 3: Now let us bring down 16, making the new dividend 316. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be 60n, where n is the next digit in the quotient. We need to find the value of n such that 60n x n ≤ 316.

Step 5: The next step is finding the largest n such that 60n x n ≤ 316. Let us consider n as 5; now, 60 x 5 x 5 = 1500, which is too large. Trying n = 4 gives 60 x 4 x 4 = 960, still too large. Trying n = 3 gives 60 x 3 x 3 = 540, which is still too large. Trying n = 2 gives 60 x 2 x 2 = 240, which is less than 316.

Step 6: Subtract 240 from 316; the difference is 76, and the quotient becomes 32.

Step 7: 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, making it 7600.

Step 8: Now, find the new divisor, which is 640 because 640 x 1 = 640, and 640 is less than 760.

Step 9: Subtracting 640 from 760 gives a result of 120.

Step 10: Now the quotient is 34.8.

Step 11: Continue doing these steps until we get sufficient decimal places.

So the square root of √1216 is approximately 34.865.

Approximation method is another method for finding 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 1216 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1216. The smallest perfect square less than 1216 is 1156, and the largest perfect square greater than 1216 is 1225. √1216 falls somewhere between 34 and 35.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using this formula, (1216 - 1156) / (1225 - 1156) = 60 / 69 ≈ 0.87. 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 34 + 0.87 = 34.87, so the square root of 1216 is approximately 34.87.

• 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.
   
• Integers: The combination of whole numbers and negative numbers are integers, for example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 are integers.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal, for example: 7.86, 8.65, and 9.42 are decimals.