Problem:
If a and b are positive numbers such that ab=ba and b=9a, then the value of a is
Answer Choices:
A. 9
B. 91β
C. 99β
D. 39β
E. 43β
Solution:
Raise both sides of ab=ba to the power 1/a and substitute b=9a to obtain a9=9a. Since aξ =0,a8=9; so a=91/8=(32)1/8=31/4.
Note. For any positive kξ =1, the equations ab=ba,b=ka have the unique positive simultaneous solution
(a,b)=(k1/(kβ1),kk/(kβ1))