The laws of indices
, , , and .
Negative and fractional indices
(a reciprocal, not a negative number); ; and .
Solve index equations by matching the base
Write both sides as powers of the same base, then equate the exponents (e.g. ).
Drawn from real examiner reports.
Multiplying the powers when you should add them
When multiplying like bases the indices ADD; students often multiply them (or vice versa), e.g. mishandling .
Not rewriting numbers to a common base
Failing to express numbers like , or as powers of , so the equation can't be solved by matching exponents.
Treating a negative index as a negative number
Thinking is negative rather than ; or forgetting the root in a fractional index.
Rewrite to a common base first
Before comparing or simplifying, turn every term into a power of the same base — then the index laws do the work.
For the same base : Multiply → add the powers. Divide → subtract. Power of a power → multiply. Mixing these up (adding when you should multiply, or the reverse) is the most penalised slip.
Simplify .
Simplify, leaving your answer as a single power of . (a) (b)
