> > > the equation t^2 = a^3 shows the relationship between a planet’s orbital period, t, and the planet’s mean distance from the sun, a, in astronomical units, au. if planet y is k times the mean distance from the sun as planet x, by what factor is the orbital period increased? < <

If planet Y is k times the mean distance from the sun than planet X, the right side of the equation becomes (kA)^3. which is k^3 times the left side, T^2. To equate both sides of the equation, multiply T by k^3/2 so that the left side becomes ((k^3/2) x T)^2 which simplifies into (k^3) x (T^2). Therefore, the answer is k^3/2.