Multiply every term inside a single bracket
To expand , multiply the outside term by each term inside: . Keep careful track of signs, e.g. . Watch the powers: , not .
Only like terms can be collected
Like terms have the same letters raised to the same powers, e.g. and , or and . Add or subtract their number parts (coefficients) and keep the letter part unchanged: . Unlike terms such as and , or and , cannot be combined.
Index laws: add, subtract, multiply powers
For the same base: (multiply → add powers), (divide → subtract), and (power of a power → multiply). Also and .
Drawn from real examiner reports.
Multiplying the powers instead of adding
When multiplying powers of the same base the indices should be added, but students often multiply them — e.g. writing instead of , or making the same slip inside an index equation.
A commonly reported examiner note records that some students were incorrectly multiplying the powers instead of adding, and a (rare) wrong answer was an incorrectly combined power.
Not using brackets when substituting
Omitting brackets changes the meaning, so at becomes (which is ) — easy to get wrong without them. Examiners explicitly advise using brackets when forming and evaluating expressions.
A commonly reported examiner note advises students to use brackets when formulating expressions or equations.
Sign slips with a negative substitution
Forgetting that a negative squared is positive, or dropping a minus sign — e.g. evaluating at as instead of , or mishandling when is negative. Writing the value in brackets, , prevents this.
Combining unlike terms
Only like terms — the same letters to the same powers — can be added or subtracted. You cannot merge with , or write as a single term, and stays as two terms. A common factor may be taken out, but that is not the same as combining them.
Doing addition before multiplication
BIDMAS still applies once letters are involved. In at , do the multiplication first: — not . Powers and multiplication are resolved before addition and subtraction.
Substitute with brackets, one step at a time
When a formula or expression is given, write the value in brackets, then apply powers and multiplications before adding or subtracting (BIDMAS). This is exactly the method examiners reward on substitution questions.
Sanity-check the size after substituting
Put the value back and check the magnitude. At , alone is , and adds a positive because is negative, so the total should exceed . An answer that comes out smaller flags a sign slip.
Expand brackets before collecting terms
Multiply out every bracket first, then gather like terms — never collect across an un-expanded bracket. For , expand to , then collect to . Watch .
aᵐ × aⁿ = ?
Simplify .
