Square Root of 1154

The square root is the inverse of the square of the number. 1154 is not a perfect square. The square root of 1154 is expressed in both radical and exponential form. In the radical form, it is expressed as √1154, whereas (1154)^(1/2) is in the exponential form. √1154 ≈ 33.962, 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 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 1154 is broken down into its prime factors:

Step 1: Finding the prime factors of 1154 Breaking it down, we get 2 x 577. The number 577 is a prime number. Therefore, 1154 = 2 x 577.

Step 2: Since 1154 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 1154 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 1154, we can group it as 11 and 54.

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, which is less than 11. Now, the quotient is 3, and after subtracting 9 from 11, the remainder is 2.

Step 3: Bring down the next group, 54, to make it 254.

Step 4: Double the quotient to get 6, which becomes the beginning of the new divisor.

Step 5: Find a digit p such that 6p x p is less than or equal to 254. In this case, p is 4 because 64 x 4 = 256, which is close to 254.

Step 6: Subtract 256 from 254 to get a remainder of -2, indicating a small error in selection, adjust by using a lower number.

Step 7: Continue with decimal values for more precision, leading to a quotient of approximately 33.96.

The approximation method is another method for finding 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 1154 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1154. The smallest perfect square less than 1154 is 1089 (33²) and the largest perfect square greater than 1154 is 1156 (34²). Thus, √1154 falls between 33 and 34.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (1154 - 1089) / (1156 - 1089) = 65 / 67 ≈ 0.97 Using the formula, we identified the decimal point of our square root. Adding this to the initial estimate gives us 33 + 0.97 = 33.97, so the square root of 1154 is approximately 33.97.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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 known as the principal square root.

• Prime factorization: The expression of a number as a product of its prime factors. For example, the prime factorization of 1154 is 2 x 577.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and subtracting iteratively to approximate the square root.