About this chapter
Algebraic expressions opens Pure 1, and its skills turn up all over the WMA11 paper. In January 2025 alone, surds and indices were needed in a straight-line question, a question on index laws and hidden quadratics, and an integration question that started by rewriting terms as powers of x.
This chapter moves from the laws of indices and factorising cubics to simplifying surds and rationalising denominators. It finishes with the ideas that separate the top grades: hidden quadratics in powers of 2, and equations solved by writing both sides as powers of the same prime. Nine of the ten questions say that every stage of working must be shown, and eight add that solutions relying on calculator technology are not acceptable.
The ten questions
- Q1 (4 marks): find the value of 27 to the power −2/3, then simplify a quotient of powers of x using the laws of indices.
- Q2 (6 marks): factorise the cubic 2x³ − 7x² − 15x completely, then expand (x+3)(2x−1)(4−x).
- Q3 (5 marks): simplify √50 − √18, then use it to rationalise a surd fraction into the form p + q√2.
- Q4 (5 marks): expand (3−2√x)², then divide by √x and write the result as a sum of powers of x.
- Q5 (6 marks): a rectangle with side (2+√3) cm and area (13+7√3) cm²: find the other side exactly, then use Pythagoras to show a result for the diagonal.
- Q6 (6 marks): let p = 2 to the power x to turn an exponential equation into the quadratic 2p² − 17p + 8 = 0, then solve the original equation.
- Q7 (6 marks): solve simultaneous index equations whose right-hand sides are 32√2 and 729√3.
- Q8 (7 marks): find the gradient between two points with surd coordinates, then where the perpendicular line crosses the x-axis, in the form a + b√3.
- Q9 (6 marks): find where the curves y = √(2x) and y = √x + 3 meet, with both coordinates exact in the form a + b√2.
- Q10 (6 marks): prove a conjugate identity, use it to rationalise a fraction, then show that a sum of 24 surd fractions telescopes to exactly 3.
Key skills tested
Laws of indices. Add powers when multiplying, subtract when dividing, multiply when raising a power to a power. For example, 6x³ ÷ 2x² = 3x.
Negative, zero and fractional indices. A negative power means one over, a power of zero gives 1, and in a fractional power the denominator is a root and the numerator is a power. For example, 8 to the power −2/3 is 1 over (cube root of 8)², which is 1/4.
Expanding brackets. Multiply two brackets first, then multiply by the third and collect like terms. For example, (x+2)(x−1)(x+3) = x³ + 4x² + x − 6.
Factorising completely. Take out the common factor first, then factorise the quadratic that is left or use the difference of two squares. For example, x³ − 9x = x(x−3)(x+3).
Writing as powers of x. Write √x as x to the power ½, then divide every term of the numerator by it. For example, (x²+3) ÷ √x becomes x to the power 3/2, plus 3x to the power −½.
Simplifying surds. Use √(ab) = √a × √b and take out the largest square factor. For example, √12 + √27 = 2√3 + 3√3 = 5√3.
Rationalising the denominator. Multiply the top and bottom by the conjugate, so the denominator becomes a difference of two squares. For example, 1/(3+√2) = (3−√2)/(9−2) = (3−√2)/7.
Hidden quadratics. Spot that 4 to the power x is the square of 2 to the power x, and let p = 2 to the power x. For example, 4 to the power x, minus 5 lots of 2 to the power x, plus 4 = 0 becomes p² − 5p + 4 = 0. So p = 1 or 4, which gives x = 0 or 2.
Equating powers. Write both sides as powers of the same prime, then set the indices equal. For example, 4 to the power x = 8√2 becomes 2 to the power 2x = 2 to the power 7/2, so x = 7/4.
Worked example
Question 1 from this chapter. (a) Find the value of 27 to the power −2/3. (b) Simplify fully 27x to the power 6, all raised to the power 2/3, divided by (3x)².
(a) A negative power means one over, so 27 to the power −2/3 is 1 over 27 to the power 2/3. In the power 2/3, the 3 is a cube root and the 2 is a square: the cube root of 27 is 3, and 3² = 9. The answer is 1/9.
(b) Apply the power 2/3 to each part of the top. 27 to the power 2/3 is 9, as in part (a), and the power 6 on x becomes 6 × 2/3 = 4, so the top is 9 times x to the power 4. The bottom is (3x)² = 9x². Dividing, the 9s cancel and the powers of x subtract, leaving x².
Both parts need every step shown, because the question says solutions relying on calculator technology are not acceptable. The full mark scheme shows where each M and A mark is awarded.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2023, Q10(b): expanding three linear brackets, the skill in Q2(b).
- January 2023, Q5, and January 2025, Q4(ii): a hidden quadratic solved by substitution, as in Q6.
- June 2023, Q4(a): splitting a fraction with roots into powers of x, as in Q4.
- January 2025, Q4(i): rewriting expressions with the laws of indices, the skill behind Q7.
- January 2025, Q2: rationalising a surd gradient, then finding a perpendicular line, as in Q8.
- June 2023, Q6: expanding, rationalising, then a harder "hence" part with surds, as in Q10.
Where marks are lost
- Stopping at p. In hidden quadratics, examiners reported many students solving for p and never going back to x. Always finish by working out x from each value of p, and reject any negative value of p, because a power of 2 can never be negative.
- Answers straight from a calculator. When a question says calculator technology is not acceptable, roots written down with no method scored nothing, and gradients that were never visibly rationalised lost marks.
- The x term in a triple expansion. The most common slip when expanding three brackets is the final x coefficient. Expand two brackets, check them, then multiply by the third, and never divide the final answer through by a number.
- Ignoring "hence". In June 2023 the final "hence" part of the surd question was rarely scored. That part is designed to use the result you have just found, so start from that result.
Common questions
Are the laws of indices in the formula booklet?
No. Nothing in this chapter is in the formula booklet, so the index laws and the conjugate method must be learned.
Can I use a calculator on surd questions?
A calculator is allowed on the paper, but surd and index questions often say that solutions relying on calculator technology are not acceptable. Then every step, including the rationalising, must be written out to earn the marks.
What does "exact" mean in an answer?
Leave surds and fractions in the answer, such as 5√3 or 1/9, rather than a rounded decimal.
Where this chapter leads
- Pure 1 Chapter 5: Straight-line Graphs: gradients and lines with surd coordinates.
- Pure 1 Chapter 8: Differentiation: rewriting roots and fractions as powers of x before differentiating.
- Pure 1 Chapter 9: Integration: the same rewriting before integrating.
- Pure 2 Chapter 3: Exponentials and Logarithms: hidden quadratics and equating powers, taken further with logarithms.