Square Root of 1.73

The square root is the inverse of the square of the number. 1.73 is not a perfect square. The square root of 1.73 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.73, whereas (1.73)^(1/2) in the exponential form. √1.73 ≈ 1.31529, 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. However, since 1.73 is not an integer, prime factorization is not applicable. For non-perfect squares and non-integers, we rely on other methods such as long-division and approximation to find the 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 take 1.73 and pair the digits from the decimal point.

Step 2: Bring down the first pair, which is 1, and find the largest number whose square is less than or equal to 1. This number is 1. Thus, the first digit of our root is 1, and the remainder is 0.

Step 3: Bring down the next pair (73), making it 173. Double the quotient (1), which becomes 2, and determine the next digit in the quotient such that 2n x n is less than or equal to 173. The closest is 6, since 26 x 6 = 156.

Step 4: Subtract 156 from 173 to get a remainder of 17.

Step 5: Since the remainder is less than the divisor, add a decimal point and bring down pairs of zeroes.

Step 6: Repeat the process with 1700, and determine the next digit in the quotient, which would be 1.315. Continue this process until the desired accuracy is achieved.

The approximation method is another method for finding 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 1.73 using the approximation method.

Step 1: Identify the perfect squares closest to 1.73. The perfect squares closest are 1 (1^2) and 4 (2^2). Thus, √1.73 is between 1 and 2.

Step 2: Use linear approximation between these two numbers. The formula is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Step 3: Applying this: (1.73 - 1) / (4 - 1) = 0.73 / 3 = 0.2433.

Step 4: Add this to 1 (the lower bound): 1 + 0.2433 ≈ 1.2433. Refinement of this estimate through further approximation or calculation will yield √1.73 ≈ 1.31529.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of this is √16 = 4.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not zero and p and q are integers.
   
• Decimal: A decimal is a number that has a whole number and a fraction part, such as 1.73.
   
• Approximation: The process of finding a value that is close enough to the correct answer, usually with some degree of accuracy.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs of digits and performing division iteratively.