Square Root of 1107

The square root is the inverse of the square of the number. 1107 is not a perfect square. The square root of 1107 is expressed in both radical and exponential form. In the radical form, it is expressed as √1107, whereas (1107)^(1/2) in the exponential form. √1107 ≈ 33.257, 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 1107 is broken down into its prime factors.

Step 1: Finding the prime factors of 1107 Breaking it down, we get 3 × 3 × 7 × 17: 3² × 7 × 17

Step 2: Now that we have found the prime factors of 1107, the second step is to make pairs of those prime factors. Since 1107 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √1107 using prime factorization 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 1107, we need to group it as 07 and 11.

Step 2: Now we need to find n whose square is less than or equal to 11. We can say n as ‘3’ because 3² = 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 07, which is the new dividend. Add the old divisor with the same number 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor becomes 6n. We need to find the value of n. Step 5: The next step is finding 6n × n ≤ 207. Let us consider n as 3. Now, 63 × 3 = 189.

Step 6: Subtract 189 from 207; the difference is 18, and the quotient is 33.

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. Now the new dividend is 1800.

Step 8: Now we need to find the new divisor. Trying 663 and multiplying by 2 (663 × 2 = 1326) fits.

Step 9: Subtracting 1326 from 1800, we get 474.

Step 10: 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 √1107 is approximately 33.257.

The approximation method is another way to find 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 1107 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1107. The smallest perfect square less than 1107 is 1089, and the largest perfect square greater than 1107 is 1156. √1107 falls somewhere between 33 and 34.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (1107 - 1089) ÷ (1156 - 1089) = 18 / 67 ≈ 0.2687 Adding this to the base of 33, we get 33 + 0.2687 ≈ 33.2687, so the square root of 1107 is approximately 33.2687.

• Square root: A square root is the inverse of a square. 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, the positive square root is often used in real-world applications, making it the principal square root.

• Prime factorization: The expression of a number as a product of prime numbers. For example, the prime factorization of 30 is 2 × 3 × 5.

• Decimal: A decimal is a number that consists of a whole number and a fractional part, separated by a decimal point. Examples include 7.86, 8.65, and 9.42.