The original base dimensions of a square pyramid are tripled while the height remains the same. how do the changes in the dimensions of the base of the pyramid affect the volume of the solid? 3 times bigger 6 times bigger 9 times bigger 27 times bigger

Well first you would just have to look at the equation for the volume of a pyramid. This is:
V = (length * width * height) / 3

and so we can just say all pyramids have a volume of V.

So now we want the base to be 3 times bigger which means we would have to multiple the length and width by 3 and the new volume equation would be
V = (3*length * 3 * width * height) / 3

we can factor the two 3's from the parenthesis and get
V = 9(l * w * h) /3

if we are looking at a ratio of how much the volume increases we can say

aV = b(l * w * h) /3

since:
V = (l * w * h) / 3

then:
aV = bV, divide both sides by V and:
a = b

using this we can see that the volume increases by a factor of 9 for 3 times bigger

now for 6 times
V = (6 * l * 6 * w * h) / 3, pull 6 * 6 out
V = 36(l * w * h) /3

and this one increases by factor of 36

if we see a pattern it always increases by the square of the factor of the growoth of the base

so for 9 times bigger it would be 9^2 = 81

and for 27 times bigger it would be 27^2 = 729