The equation given above is wrongly balanced, so I have to correct it first.
Al^@Al∘ + Zn(NO_3)_2Zn(NO3)2 = Al(NO_3)_3Al(NO3)3 + Zn^@Zn∘ (unbalanced)
Tallying the atoms based on subscripts,
left side: AlAl = 1; ZnZn = 1; (NO_3)_3(NO3)3 = 2
right side: AlAl = 1; ZnZn = 1; (NO_3)_3(NO3)3 = 3
Notice that I am considering the NO_3^-NO−3 ion as one "atom" in order to avoid confusing myself.
Balancing the equations we have:
color (blue) 2Al^@ (s) 2Al∘(s) + color (red) 3 Zn(NO_3)_23Zn(NO3)2 = color (green) 2Al(NO_3)_32Al(NO3)3 + color (magenta) 3Zn^@ (s)3Zn∘(s) (balanced)
left side: AlAl = (1 x color (blue) 22) = 2; ZnZn = (1 x color (red) 33) = 3; (NO_3)_3(NO3)3 = (2 x color (red) 33) = 6
right side: AlAl = (1x color (green) 22) = 2; ZnZn = (1 x color (magenta) 33) = 3; (NO_3)_3(NO3)3 = (3 x color (green) 22) = 6
Now that we have the correct balance equation, we must rewrite the equation showing all the ions involved in the reaction.
2Al^@ (s) 2Al∘(s) + 3Zn^"2+"3Zn2+ + 6NO_3^-6NO−3 = 2Al^"3+"2Al3+ + 6NO_3^-6NO−3 + 3Zn^@ (s)3Zn∘(s)
Notice that for the NO_3^-NO−3, I multiply the subscript with the substance's coefficient. Now let us get rid of your spectator ions (ions that are present in both sides of the equation).
2Al^@ (s) 2Al∘(s) + 3Zn^"2+"3Zn2+ + cancel (6NO_3^-) = 2Al^"3+" + cancel (6NO_3^-) + 3Zn^@ (s)
Now let's show the electrons per half-reaction:
2Al^@ (s) = 2Al^"3+" + 6e^-
3Zn^"2+" + 6e^- = 3Zn^@ (s)
Thus, the net ionic equation is
2Al^@ (s) + 3Zn^"2+" = 2Al^"3+" + 3Zn^@ (s)