Is a positive? (1) a^6 > a (2) a^5 > a

Number line

If you want to make use of number properties

A) a>1….As power increases, value of a increases. => \(a^3>a^2>a\)

B) 0<a<1….As power increases, value of a decreases. => \(a^3<a^2<a\)

C) -1<a<0….As odd power increases, value of a increases, but as even power increases, the value of a decreases but remains more than odd power => \(a^2>a^4>a^3>a\)

D) a<-1….As odd power increases, value of a decreases, but as even power increases, the value of a increases and remains more than odd power always => \(a^4>a^2>a>a^3>a^5\)

(1) \(a^6 >a\)
Cases A, B and D
Insufficient

(2) \(a^5 >a\)
Cases A and D
Insufficient

Combined
Cases A and D
Both when a>1, and when a<-1, the statements are true. Thus a can be both positive and negative.
Insufficient

____________________________________xx________________________________________________________
Algebraic

(1) \(a^6 >a…….a(a^5-1)>0\)
When a>0, then a^5>1……..a>1 or a is positive
When a<0, then a^5<1……..a<0 or a is negative
Insufficient

(2) \(a^5 >a……..a(a^4-1)>0\)
When a>0, then a^4>1……..a>1 or a is positive
When a<0, then a^4<1……..-1<a<0 or a is negative
Insufficient

Combined
Both when a>1, and when -1<a<0, the statements are true. Thus a can be both positive and negative.
Insufficient


E