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. 91​
C. 99​
D. 39​
E. 43​
Solution:
Raise both sides of ab=ba to the power 1/a and substitute b=9a to obtain a9=9a. Since aî€ =0,a8=9; so a=91/8=(32)1/8=31/4.
Note. For any positive kî€ =1, the equations ab=ba,b=ka have the unique positive simultaneous solution
(a,b)=(k1/(k−1),kk/(k−1))