How many years would it take the planet to orbit the sun. How do you calculate. All I have is, that the planet is 97 times farther from sun than earth or 97 Astronomical units

Question

Using Kepler's Law you find that the anser is 97^(1.5) years. You can estimate this without using a calculator:

97^(1.5) = (100-3)^(1.5) =
[100*(1 - 3/100)]^(1.5) =
100^(1.5)*(1-3/100)^(1.5)=

1000*[1-1.5*3/100 + (1.5)*(0.5)/2 (3/100)^2 + term of order (3/100)^3] =

1000 - 45 + 27/80 + term of order 1/100

27/80 is about 1/3, so the answer is
955.3 years.