Square Root of 1189

The square root is the inverse of the square of the number. 1189 is not a perfect square. The square root of 1189 is expressed in both radical and exponential form. In radical form, it is expressed as √1189, whereas in exponential form it is (1189)^(1/2). √1189 ≈ 34.49334, 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 like 1189, the prime factorization method is not applicable. Instead, 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 1189 is broken down into its prime factors.

Step 1: Finding the prime factors of 1189 Breaking it down, we get 1189 = 13 x 91 = 13 x 7 x 13: 13^2 x 7

Step 2: We found the prime factors of 1189. However, since 1189 is not a perfect square, the digits of the number cannot be grouped into pairs.

Therefore, calculating √1189 using prime factorization might not be 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 1189, we group it as 89 and 11.

Step 2: Now, we need to find n whose square is less than or equal to the leftmost group. For 11, n is 3 because 3 x 3 = 9 which is less than 11. The quotient is 3 and the remainder is 11 - 9 = 2.

Step 3: Bring down the next pair 89, making the new dividend 289. Add the old divisor with the same number 3 + 3 to get 6 as the new divisor.

Step 4: Find n such that 6n x n is less than or equal to 289. Let n be 4, then 64 x 4 = 256.

Step 5: Subtract 256 from 289; the difference is 33, and the quotient is 34.

Step 6: Since the dividend is less than the divisor, add a decimal point to the quotient. Adding the decimal point allows us to bring down two zeroes, making the new dividend 3300.

Step 7: Find a new divisor. The divisor is now 34 + 34 = 68. Find n such that 68n x n ≤ 3300. Let n be 4, then 684 x 4 = 2736.

Step 8: Subtract 2736 from 3300, resulting in 564.

Step 9: Bring down another pair of zeroes, making the dividend 56400.

Step 10: Continue this process.

The square root of 1189 is approximately 34.49.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 1189 using the approximation method.

Step 1: Find the closest perfect squares around 1189. The closest perfect squares are 1156 (34^2) and 1225 (35^2). √1189 falls between 34 and 35.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (1189 - 1156) / (1225 - 1156) = 33/69 ≈ 0.478 Add this value to 34 (the square root of the nearest smaller perfect square): 34 + 0.478 ≈ 34.478

So, the square root of 1189 is approximately 34.478.

• 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.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 144 is a perfect square because it is 12^2.
   
• Long division method: A method used to find the square root of numbers that are not perfect squares. It involves dividing the number into groups and performing a series of divisions.
   
• Approximation method: A method used to estimate the square root of a number by finding the closest perfect squares and using a formula to approximate the square root.