Step 1 :The problem is asking for the time it takes to fill the pool when both the pipe and the hose are used simultaneously. This is a problem of rates. The rate of the pipe is 1 pool per 13 hours and the rate of the hose is 1 pool per 21 hours. The combined rate is the sum of the individual rates. The time it takes to fill the pool is the reciprocal of the combined rate.
Step 2 :First, calculate the rate of the pipe, which is \(\frac{1}{13}\) pools per hour, approximately 0.07692307692307693 pools per hour.
Step 3 :Next, calculate the rate of the hose, which is \(\frac{1}{21}\) pools per hour, approximately 0.047619047619047616 pools per hour.
Step 4 :Then, calculate the combined rate by adding the rate of the pipe and the hose, which is approximately 0.12454212454212454 pools per hour.
Step 5 :Finally, calculate the time it takes to fill the pool by taking the reciprocal of the combined rate, which is approximately 8.029411764705882 hours.
Step 6 :Final Answer: The time it will take to fill the pool if water is entering through both the pipe and the hose is approximately \(\boxed{8.03}\) hours.