Square Root of 273

The square root is the inverse of squaring a number. 273 is not a perfect square. The square root of 273 is expressed in both radical and exponential form. In radical form, it is expressed as √273, whereas in exponential form it is (273)^(1/2). √273 ≈ 16.5227, 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 273, the long-division method and approximation method are more suitable. Let us explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Let's look at how 273 is broken down into its prime factors:

Step 1: Finding the prime factors of 273 Breaking it down, we get 3 x 7 x 13.

Step 2: Now that we have found the prime factors of 273, the next step is to make pairs of those prime factors. Since 273 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 273 using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect square numbers. In this method, we find the closest perfect square number for the given number. Let us learn how to find the square root using the long division method, step by step:

Step 1: To begin with, group the numbers from right to left. In the case of 273, we group it as 73 and 2.

Step 2: Now, find a number n whose square is less than or equal to 2. We can say n is ‘1’ because 1 x 1 is less than or equal to 2. The quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 73, making the new dividend 173. Add the previous divisor with the same number: 1 + 1 = 2, which will be our new divisor.

Step 4: The new divisor is now 2n. We need to find the value of n such that 2n x n ≤ 173. Let us consider n as 6, now 26 x 6 = 156.

Step 5: Subtract 156 from 173; the difference is 17, and the quotient is 16.

Step 6: Since the dividend is less than the divisor, we add a decimal point and two zeroes to the remainder. The new dividend is 1700.

Step 7: Find a new divisor. The new divisor is 326, and we find n as 5 because 326 x 5 = 1630.

Step 8: Subtract 1630 from 1700, we get the result 70.

Step 9: The quotient is now 16.5.

Step 10: Continue doing these steps until you get two decimal places. If there are no decimal values, continue until the remainder is zero.

So, the square root of √273 ≈ 16.52.

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 273 using the approximation method.

Step 1: Find the closest perfect squares to √273. The smallest perfect square below 273 is 256, and the largest perfect square above 273 is 289. √273 falls somewhere between 16 and 17.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (273 - 256) / (289 - 256) = 17/33 ≈ 0.515.

Step 3: Add the decimal approximation to the lower perfect square: 16 + 0.515 ≈ 16.515. Therefore, the square root of 273 is approximately 16.52.

• Square root: The square root is the inverse of squaring a number. Example: 4² = 16, and the inverse is √16 = 4.

• Irrational number: An irrational number cannot be written as a simple fraction; it has non-repeating, non-terminating decimals.

• Perfect square: A perfect square is a number that can be expressed as the square of an integer.

• Long division method: A method used to find the square root of non-perfect squares by using division.

• Approximation: A method to find an approximate value of square roots by comparing them to nearby perfect squares.