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Challenge: indices + factorisation combo

2026-11-01Exam & ChallengeMs Chu

Harder multi-step items linking index laws with expansion and factorisation.

Key formulas

Simplify indices before expanding when helpful\text{Simplify indices before expanding when helpful}
Factor completely:numbers and brackets\text{Factor completely}: \text{numbers and brackets}

Why combine skills

Harder school exam questions stack skills: simplify an index expression, then factor the result, or expand and then apply difference of squares.

Keep structure visible

Rewrite 9x² − 4 as (3x)² − 2² before factoring. Rewrite 16 as 2⁴ or 4² depending on the base you need.

Positive indices at the end

If the question demands positive indices, finish with that rewrite even after factorisation.

Exam habit

Plan the route in five words (“index → expand → factor”) before calculating.

Worked example

Difference of squares

Problem. Expand (3x − 2)(3x + 2).

Solution. 9x² − 4, which is already (3x−2)(3x+2) when factored back.

Review quiz

1 / 4

(3x−2)(3x+2) =

Practice questions

  1. Simplify (2^5 × 2^{−2}) ÷ 2 and factorise the numerical answer into primes.
  2. Expand (3x − 2)(3x + 2) and factorise your answer if possible.
  3. Simplify (x^3 y^{−1})^2 / (x y^{−3}) with positive indices.
  4. Factorise 18x² − 8 completely.

Study notes

Is 9x² − 4 prime?

No — it factors as (3x−2)(3x+2).

When do I stop factorising?

When no further common factors or standard patterns remain over the integers.

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