38 ants, including the queen, created a new colony. after 9 months, there are now 970 ants. at 17 months, how many ants will live in the colony? (show your work)

At 17 months 17,275 ants will live in the colony

Number of months since the new colony was created: x
Number of ants: y

y=a(b)^x

At the beginning 38 ants, including the queen, created a new colony:
x=0→y=38:
38=a(b)^0
38=a(1)
38=a
a=38

y=38(b)^x

After 9 months there are now 970 ants:
x=9→y=970
970=38(b)^9
Solving for b. Dividing both sides by 38:
970/38=38(b)^9/38
25.52631578=b^9
ninth root:
ninth root (25.52631578)=ninth root (b^9)
1.433283216=b
b=1.433283216

y=38 (1.433283216)^x

At 17 months, how many ants will live in the colony?
x=17, y=?
y=38 (1.433283216)^17
y=38 (454.6155221)
y=17,275.38984
y=17,275