To find out how many days it takes for both A and B to complete the work together, we can use the concept of work rates. Here’s how it can be done:

1. Let’s denote the total work as $W$.

2. A can complete the work in 10 days. So, A’s work rate is $\frac{W}{10}$ per day.

3. B can complete the work in 8 days. So, B’s work rate is $\frac{W}{8}$ per day.

4. When A and B work together, their combined work rate is the sum of their individual work rates: $\text{Combined work rate} = \frac{W}{10} + \frac{W}{8}$

5. To add these fractions, we need a common denominator. The least common multiple of 10 and 8 is 40. So, $\frac{W}{10} = \frac{4W}{40} \quad \text{and} \quad \frac{W}{8} = \frac{5W}{40}$

6. Adding these fractions gives: $\frac{4W}{40} + \frac{5W}{40} = \frac{9W}{40}$

7. The combined work rate of A and B is $\frac{9W}{40}$ per day. Therefore, the number of days it takes for A and B to complete the work together is: $\text{Time} = \frac{W}{\frac{9W}{40}} = \frac{40}{9} \text{ days}$

So, A and B together can complete the work in $\frac{40}{9} \approx 4.44$ days.

Would you like a detailed breakdown or have any questions? Here are some other questions you might be interested in:

1. How to solve quadratic equations?
2. How to find the derivative of a function?
3. What is the Pythagorean theorem?
4. How to integrate a function?
5. What are the properties of logarithms?

Tip: When dealing with work rate problems, always convert the individual rates into a common unit before combining them.