D and E are eliminated because 'their' has an unclear reference.

A and C are eliminated because both refers to the chips and devices, not the power.

D

Another way to look at it is parallelism: Both X and Y. In B , X=chips;Y= Electronic device whereas in A and C, X= The power of chips(phrase);Y= the electronic device(Noun). Hence B.

Another way to look at it is parallelism: Both X and Y. In B , X=chips;Y= Electronic device whereas in A and C, X= The power of chips(phrase);Y= the electronic device(Noun). Hence B.

As the etched lines on computer memory chips have become thinner and the chips' circuits more complex, both the power of the chips and the electronic devices they drive have vastly increased.

A. the chips' circuits more complex, both the power of the chips and the electronic devices they drive have -- incorrect parallelism (both X and Y)here x= phrase, Y=noun
B. the chips' circuits more complex, the power of both the chips and the electronic devices they drive has-- correct
C. the chips' circuits are more complex, both the power of the chips and the electronic devices they drive has-- incorrect parallelism (both X and Y)here x= phrase, Y=noun and verb used is incorrect
D. their circuits are more complex, the power of both the chips and the electronic devices they drive have -- unclear referent for their and verb should be has
E. their circuits more complex, both the power of the chips and the electronic devices they drive have -- unclear referent for their and incorrect parallelism

Correct me if I am wrong