Question

Diophantus's methods can be used to approximate √(17) by solving the following problem: Add a small square to 17 to obtain a square. Use Diophantus's method to approximate √(17) to three decimal places.

Answer 1

To approximate √17 using Diophantus's method, find a small square to add to 17 and then continue using the square root formula with the new value to approximate the square root to the desired level of precision.

Step-by-step explanation:

To approximate √17 using Diophantus's method, we first need to find a small square that we can add to 17 to obtain a square. Let's start by using the formula √a = a1/2. In this case, a = 17, so √17 = 171/2. To find a small square to add to 17, we need to find a perfect square close to 17. The perfect square closest to 17 is 16, which is 42. We can then add this small square to 17: 17 + 4 = 21. Now we can find the square root of 21 using Diophantus's method. We start by using the formula √a = a1/2. In this case, a = 21, so √21 = 211/2. We continue this process until we reach the desired level of precision.