Final answer:
To approximate the square root of 15, we can use long division to find a rational number that is close to the square root. The rational approximation of the square root of 15 is approximately 3.87.
Step-by-step explanation:
To find the rational approximation of the square root of 15, we can start by finding a rational number that is close to the square root of 15. One way to do this is by prime factorizing 15: 15 = 3 * 5. Since neither 3 nor 5 are perfect squares, we can use a technique called long division to approximate the square root of 15.
Let's start with an initial approximation of 3. Divide 15 by 3 to get 5. The average of 3 and 5 is 4. This means that the square root of 15 is closer to 4 than it is to 3. Repeat the process with 4 as the new approximation. Divide 15 by 4 to get 3.75. The average of 4 and 3.75 is 3.875. Continue this process until the desired level of accuracy is achieved.
The rational approximation of the square root of 15 is approximately 3.87.