Using the change in oxidation number method, balance the redox reaction shown below.

(NO2^2)+ (MnO4^-)---> (NO3^-)+ Mn^2+

Divide into half reactions. I will show you how to do MnO4^- and you do the other one.
MnO4^- ==> Mn^+2

Step 2. Determine the oxidation state of the element changing. In this case, Mn changes from +7 on the left to +2 on the right.

Step 3. Add electrons to the appropriate side to balance the change in oxidation state.
MnO4^- + 5e ==> Mn^+2

Step 4. Count the charge on each side and add H^+ (in an acid solution) or OH^- (in a basic solution) to balance the charge.
I see -6 on the left and +2 on the right. This is an acid solution, therefore, add 8H^+ to the left.
MnO4^- + 8H^+ + 5e ==> Mn^+2

Step 5. Add H2O (usually to the other side) to balance H and O.
MnO4^- + 8H^+ + 5e ==> Mn^+2 + 4H2O

Step 6. Check everything.

Post specific questions if you have them.

Sure! Let's balance the other half reaction:

Step 2. Determine the oxidation state of the element changing. In this case, nitrogen (N) changes from +2 on the left to +5 on the right.

Step 3. Add electrons to the appropriate side to balance the change in oxidation state.
(NO2^2+) + e ==> (NO3^-)

Step 4. Count the charge on each side and add H^+ (in an acid solution) or OH^- (in a basic solution) to balance the charge.
I see 0 on the left and -1 on the right. This is an acid solution, therefore, add 1H^+ to the right side.
(NO2^2+) + e ==> (NO3^-) + H^+

Step 5. Add H2O (usually to the other side) to balance H and O.
(NO2^2+) + e ==> (NO3^-) + H2O

Step 6. Check everything.

The balanced half-reactions are:
MnO4^- + 8H^+ + 5e ==> Mn^+2 + 4H2O
(NO2^2+) + e ==> (NO3^-) + H2O

Now, combine the two half-reactions and ensure that the number of electrons is equal in both half-reactions. In this case, multiply the second half-reaction by 5 to equalize the number of electrons.

5(NO2^2+) + 5e ==> 5(NO3^-) + 5H2O

Now, add the two half-reactions together:

MnO4^- + 8H^+ + 5(NO2^2+) + 5e ==> Mn^+2 + 4H2O + 5(NO3^-) + 5H2O

Simplify:

MnO4^- + 8H^+ + 5(NO2^2+) ==> Mn^+2 + 9H2O + 5(NO3^-)

And that's the balanced redox reaction using the change in oxidation number method!