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.
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 .
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 them
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.
Paraphrased from Nov 2024 Paper 1H (Q3a) and Paper 2H (Q14): on 2H "others were incorrectly multiplying the powers instead of adding", and on 1H the (rare) wrong answer was an incorrectly combined power.
Sign slips substituting a negative
Forgetting that a negative number squared is positive, or dropping a minus sign, e.g. evaluating at as instead of , or mishandling when is negative. Putting the substituted value in brackets, , prevents this.
Not using brackets when substituting
Omitting brackets changes the meaning, so at becomes (which is ) — easy to get wrong without brackets. Examiners explicitly advise using brackets when forming and evaluating expressions.
Paraphrased from the Nov 2024 Paper 1H report's closing advice to students to "use brackets when formulating expressions or equations".
Adding coefficients vs adding exponents
Collecting like terms adds COEFFICIENTS (); multiplying powers adds EXPONENTS (). Do not blur them: , and .
A negative outside a bracket flips every sign
A negative term outside a bracket flips EVERY sign inside: , not . The last term becomes because a negative times a negative is positive.
Substitute in brackets, then use BIDMAS
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 for substitution questions, e.g. into a volume formula.
Check a simplification by putting in a number
Test a simplified expression by substituting a simple value (like ) into both the original and your answer — they must give the same total. This catches sign slips and dropped terms in seconds.
Name the operation before touching powers
With powers of the same base, name the operation first: multiplying ADD the indices, dividing SUBTRACT, power of a power MULTIPLY. Writing which one applies stops the "multiply instead of add" slip.
Arrow the bracket to every inside term
When expanding, draw an arrow from the outside term to EACH term inside so none is missed, and multiply letters as well as numbers. Only collect like terms once every bracket has been expanded.
What does it mean for two terms to be "like terms"?
Simplify .
