Square Root of 1108

The square root is the inverse of the square of the number. 1108 is not a perfect square. The square root of 1108 is expressed in both radical and exponential form. In the radical form, it is expressed as √1108, whereas (1108)^(1/2) in the exponential form. √1108 ≈ 33.2762, 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 squares like 1108, methods such as the long-division method and approximation method are used. Let us now learn these 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 1108 is broken down into its prime factors:

Step 1: Finding the prime factors of 1108 Breaking it down, we get 2 x 2 x 277: 2² x 277

Step 2: Now we have found the prime factors of 1108. Since 1108 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 1108 using prime factorization alone is inadequate for finding an exact square root.

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 1108, we need to group it as 08 and 11.

Step 2: Now we need to find n whose square is less than or equal to 11. We can say n is ‘3’ because 3 x 3 = 9 is less than 11. Now the quotient is 3; after subtracting 9 from 11, the remainder is 2.

Step 3: Now let us bring down 08, making the new dividend 208. 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 in the form of 6n. We need to find the value of n where 6n x n ≤ 208. Let us consider n as 3, now 63 x 3 = 189.

Step 5: Subtract 189 from 208. The difference is 19, and the quotient becomes 33.

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 1900.

Step 7: Now we need to find the new divisor. Trying with 33, we find 666 x 3 = 1998 is too large, so we reduce it.

Step 8: Using 32 with 665, we find 665 x 2 = 1330.

Step 9: Subtracting 1330 from 1900, we get the result 570.

Step 10: Now the quotient is 33.2.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue until the remainder is zero.

So the square root of √1108 ≈ 33.28.

The approximation method is another method for finding the 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 1108 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1108. The smallest perfect square below 1108 is 1024, and the largest perfect square above 1108 is 1156. √1108 falls somewhere between 32 and 34.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Going by the formula (1108 - 1024) ÷ (1156 - 1024) = 84 ÷ 132 ≈ 0.6363 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 32 + 0.64 ≈ 32.64.

So, the square root of 1108 is approximately 32.64.

• Square root: A square root is the inverse of a square. For example, 4² = 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. This is why it is also known as the principal square root.

• Approximation: Approximating is finding a value close to the actual answer. In square roots, it refers to estimating a non-perfect square root.

• Long division method: A systematic method for finding the square root of numbers, especially those that are not perfect squares, which involves dividing, multiplying, and subtracting in a sequence of steps.