Square Root of 173

The square root is the inverse operation of squaring a number. 173 is not a perfect square. The square root of 173 is expressed in both radical and exponential form. In radical form, it is expressed as √173, whereas in exponential form it is expressed as (173)^(1/2). √173 ≈ 13.152, 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 typically used for perfect square numbers. However, for non-perfect square numbers like 173, the long division method and approximation method are used. Let's learn about the following methods: 

• Prime factorization method
  
   
• Long division method
  
   
• Approximation method

Prime factorization involves expressing a number as a product of its prime factors.

However, 173 is a prime number itself, and there are no repeated prime factors to pair.

Therefore, calculating √173 using the prime factorization method is not applicable.

The long division method is used for non-perfect square numbers. This method involves finding pairs of digits and then estimating the square root step by step.

Step 1: Start by pairing the digits from right to left. For 173, pair as 1 and 73.

Step 2: Find the largest number whose square is less than or equal to 1. This number is 1. Subtract 1 from 1, leaving a remainder of 0.

Step 3: Bring down the next pair of digits, 73, forming the new dividend.

Step 4: Double the quotient (1), giving a new divisor of 2. Find a number n such that 2n × n ≤ 73. The number is 3 since 23 × 3 = 69.

Step 5: Subtract 69 from 73, which results in a difference of 4.

Step 6: Since the remainder is less than the divisor, add a decimal point and bring down two zeros to get 400.

Step 7: Double the current quotient (13) to get 26. Find the number n such that 26n × n ≤ 400. The number is 1 since 261 × 1 = 261.

Step 8: Subtract 261 from 400, which leaves a remainder of 139.

Step 9: Continue this process to get more decimal places until the desired precision. The result is that √173 ≈ 13.152.

The approximation method provides a quick way to estimate square roots. Let's find the square root of 173 using this method.

Step 1: Identify the closest perfect squares around 173. These are 169 (13²) and 196 (14²). √173 lies between 13 and 14.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Applying the formula: (173 - 169) / (196 - 169) = 4 / 27 ≈ 0.148 Add this decimal to the smaller root: 13 + 0.148 = 13.148

Therefore, √173 ≈ 13.148.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16, and the inverse operation is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. For example, √173 is irrational because it cannot be written as p/q, where p and q are integers.
   
• Prime number: A prime number has only two factors: 1 and itself. For example, 173 is a prime number.
   
• Decimal: A decimal is a number that includes a whole number and a fractional part. For example, 13.152 is a decimal.
   
• Long division method: A method used to find square roots of non-perfect squares by dividing and estimating step by step.