S2 · Algebra · ALG.POWER.LAWS

Laws of exponents

Intuition

Understand first, compress later.

Algebra describes structures generally: a symbol stands for a number, while the rules remain the same.

Switch representations

Arithmetic sequencea₁ = 2, d = 3

terms aₙstep +3

A sequence is defined only for the natural numbers n = 1, 2, 3, … — which is why this plate shows separate dots on stems rather than a continuous curve; there is no term between two dots. Each step adds 3 (the recurrence), while the closed form gives any term directly: a_n = 2 + (n − 1)·3 = 3n − 1.

Formal view

You can laws of exponents when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.