I am doing this exercise for the GMAT test
Which of the following option is closest to $49^{4}81^{5}$?
A. $8^{18}$
B. $8^{19}$
C. $8^{20}$
D. $8^{21}$
E. $8^{22}$
My attempt:
$49^{4}81^{5} = (7^2)^4(9^2)^5 = 7^{8}9^{10} = (8-1)^{8}(8+1)^{10} \approx 8^{8}8^{10} = 8^{18}$
But I am not sure how $(8-1)^{8}(8+1)^{10}$ is closer to $8^{18}$ than to $8^{19}$.
Please shed me some lights. Thank you for your help!
Let's take it exactly for a few more steps: $$ (8-1)^{8}(8+1)^{10} =(8-1)^8(8+1)^8(8+1)^2\\ =(8^2-1)^8\cdot9^2<8^{16}\cdot 9^2 $$ And $9^2$ is much closer to $8^2$ than to $8^3$.