In this paper we prove the functoriality of the exterior square of cusp forms on
G
L
4
GL_{4}
as automorphic forms on
G
L
6
GL_{6}
and the symmetric fourth of cusp forms on
G
L
2
GL_{2}
as automorphic forms on
G
L
5
GL_{5}
. We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain
L
L
-functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of
G
L
4
GL_{4}
and the absolute convergence of the exterior square
L
L
-functions of
G
L
4
GL_{4}
. Second, we prove that the fourth symmetric power
L
L
-functions of cuspidal representations of
G
L
2
GL_{2}
are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound
3
26
\frac {3}{26}
for Hecke eigenvalues of Maass forms over any number field.