The index laws, used fluently
All three laws should now run together inside a single longer simplification.
By Stage 9 the three laws of indices should be second nature. The new skill is using them together — a single expression often needs more than one law before it is fully simplified.
The three laws, all needing the same base:
- Multiplying: — add the indices.
- Dividing: — subtract the indices.
- Power of a power: — multiply the indices.
Take . Multiply on the top first: . Then divide: . Naming which law you are using at each step keeps a longer simplification under control.
- Multiplying powers of the same base adds the indices.
- Dividing powers of the same base subtracts the indices.
- A power of a power multiplies the indices.
- Longer expressions often need two or three laws in turn.