TERM 1 · ALGEBRAIC EXPRESSIONS & EXPONENTSEXPRESSIONS & EXPONENTS
What you must be able to do
Classify real numbers as rational or irrational, place a simple surd between two integers, and round sensibly.
Multiply a binomial by a trinomial.
Factorise fully, including trinomials, grouping in pairs and the sum and difference of two cubes.
Simplify algebraic fractions by factorising first.
Apply the exponent laws with integer and rational exponents, and solve a simple equation with a rational exponent.
Core content · Real numbers, surds and rounding
📖 KEY DEFINITIONRational: can be written as a/b, with a and b integers and b ≠ 0; the decimal stops or repeats. Irrational: cannot; the decimal never stops and never repeats. A surd is irrational unless the number under the root is a perfect square (for ∛, a perfect cube).
Classify before you calculate
Number
Type
Why
0,375
rational
stops: 375/1000 = 3/8
0,2̇7̇
rational
repeats: equals 3/11 (below)
√49
rational
perfect square: √49 = 7
∛16
irrational
16 is not a perfect cube
22/7
rational
a fraction, only an approximation of π
π
irrational
never stops, never repeats
Worked example · Show that 0,2̇7̇ is rational. (3)
✓ A: Let x = 0,2727…; two digits repeat, so 100x = 27,2727…
✓ A: 100x − x = 27, so 99x = 27
✓ A: x = 27/99 = 3/11, a ratio of integers, so it is rational
Multiply by 10k, where k is the number of repeating digits.
Worked example · Without a calculator, determine between which two consecutive integers √52 lies. (2)
✓ A: Perfect squares either side: 49 < 52 < 64, so √49 < √52 < √64
✓ A: 7 < √52 < 8
For a cube root use cubes: 8 < 20 < 27 gives 2 < ∛20 < 3. A bare calculator value scores nothing here.
Worked example · (a) Calculate π√7 correct to two decimal places. (b) 213 learners travel in 60-seater buses. How many buses are needed? (3)
⚠️ COMMON MISTAKERounding too early. Rounding √7 to 2,65 and π to 3,14 first gives 2,65 × 3,14 = 8,321 ≈ 8,32, and the accuracy mark goes. Keep the full value; round once, on the last line.
The first product collapses to x³ − 27: a difference of cubes run backwards.
⚠️ COMMON MISTAKEThe minus before the second product flips the sign of every term in it. Keep that product in brackets until it is fully expanded. Writing x³ − 27 − x³ − x² − 3x + 2 gives −x² − 3x − 25, and the answer mark is lost.
💡 EASY MARKTen-second check: put x = 1 into the question and your answer. (−2)(13) − (−1)(1) = −25 and 1 + 3 − 29 = −25. Different values prove an error.
Core content · Factorising fully
Fully means no bracket breaks down further. Work in a fixed order:
Take out the highest common factor.
Count the terms. Two: difference of squares or cubes. Three: trinomial. Four: group in pairs.
Test every bracket again.
⚖️ LAWS / FRAMEWORKSTwo cubes a³ + b³ = (a + b)(a² − ab + b²) a³ − b³ = (a − b)(a² + ab + b²) SOAP: binomial Same sign as the question, middle term Opposite, last term Always Positive.
Worked example · Factorise fully: 3x³ − 24 (3)
✓ A: Common factor first: 3(x³ − 8)
✓ A: x³ − 2³ is a difference of cubes: a = x, b = 2
✓ A: 3(x − 2)(x² + 2x + 4)
Worked example · Factorise: 6x² − x − 15 (2)
✓ A: ac = −90; −10 and 9 add to −1: 6x² − 10x + 9x − 15 = 2x(3x − 5) + 3(3x − 5)
✓ A: = (3x − 5)(2x + 3)
Inspection earns the same marks. Check the middle term: −10x + 9x = −x.
Worked example · Factorise fully: x³ − 2x² − 9x + 18 (3)
✓ A: Group in pairs: x²(x − 2) − 9(x − 2)
✓ A: (x − 2)(x² − 9)
✓ A: Difference of two squares: (x − 2)(x − 3)(x + 3)
Taking out −9 turns +18 into −2; the brackets must match before one comes out.
⚠️ COMMON MISTAKEStopping one step short. (x − 2)(x² − 9) is not fully factorised, and the last mark goes. The opposite trap: the trinomial from two cubes (x² + 2x + 4) does not factorise, and nor does a sum of squares such as x² + 4.
Core content · Algebraic fractions
Factorise every numerator and denominator, then cancel common factors. To divide, multiply by the reciprocal. To add or subtract, use the lowest common denominator (LCD) built from the factors. The hardest Grade 10 denominator is a sum or difference of cubes.
Worked example · Simplify fully: (x³ + 8)/(x² − x − 6) × (x² − 9)/(x² − 2x + 4) (4)
Worked example · Simplify: 1/(x − 1) − 3/(x³ − 1) (4)
✓ A: x³ − 1 = (x − 1)(x² + x + 1), the LCD
✓ A: = (x² + x + 1 − 3) / [(x − 1)(x² + x + 1)]
✓ A: = (x + 2)(x − 1) / [(x − 1)(x² + x + 1)]
✓ CA: = (x + 2)/(x² + x + 1)
Always try to factorise the new numerator: that last cancel is the mark most answers miss.
⚠️ COMMON MISTAKECancelling terms, not factors. (x + 6)/(x + 3) is not 2, and (2x + 6)/2 is x + 3, not x + 6. Cancel only what multiplies the whole numerator and denominator.
Core content · Laws of exponents
The Grade 9 laws hold for x, y > 0 and integers m, n, and they act on multiplication and division only.
The six laws, revised from Grade 9
Law
Rule
Example
Product
xm · xn = xm+n
a³ · a⁴ = a⁷
Quotient
xm ÷ xn = xm−n
a⁹ ÷ a² = a⁷
Power of a power
(xm)n = xmn
(a²)⁵ = a¹⁰
Power of a product
(xy)m = xmym
(2a)³ = 8a³
Zero
x⁰ = 1, x ≠ 0
5a⁰ = 5
Negative
x−n = 1/xn, x ≠ 0
2⁻³ = 1/8
Worked example · Simplify fully, with positive exponents: (2x³y)² × 3x⁻⁴ / (6x²y⁵) (3)
✓ A: (2x³y)² = 4x⁶y²
✓ A: 4x⁶y² × 3x⁻⁴ = 12x²y²
✓ CA: 12x²y² ÷ 6x²y⁵ = 2y⁻³ = 2/y³
⚠️ COMMON MISTAKENo law works across + or −. 3x + 3x = 2 · 3x, not 32x, and 2³ + 2⁴ = 24, not 2⁷. Split the powers and take out a common factor.
Worked example · Simplify without a calculator. (4)
(3x+2 − 3x) ÷ (4 · 3x−1)
✓ A: 3x+2 = 3x · 9 and 3x−1 = 3x · ⅓
✓ A: Numerator: 3x(9 − 1) = 8 · 3x
✓ A: = 8 · 3x ÷ (4 · ⅓ · 3x)
✓ CA: = 8 × ¾ = 6
Worked example · Simplify fully: (12ⁿ · 9ⁿ⁺¹) / (27ⁿ · 4ⁿ⁻¹) (4)
✓ A: Prime bases: 12n = 22n · 3n; 9n+1 = 32n+2
✓ A: 27n = 33n; 4n−1 = 22n−2
✓ A: 22n−(2n−2) · 3n+2n+2−3n = 2² · 3²
✓ CA: = 36
Different bases built from the same primes: rewrite in primes, then collect prime by prime.
Core content · Simplifying with rational exponents
Grade 10 accepts that the six laws also hold for rational m and n. A fractional exponent is a root: xm/n = (ⁿ√x)m for x > 0.
⚡ MUST MEMORISEDenominator = root, numerator = power. Know on sight: 2⁵ = 32, 2⁶ = 64, 3⁴ = 81, 5³ = 125, 6³ = 216, and the squares to 15² = 225.
Worked example · Simplify without a calculator. (3)
82/3 + 81−1/4
✓ A: Root first: 82/3 = (∛8)² = 4
✓ A: 81−1/4 = 1/811/4 = ⅓
✓ CA: 4 + ⅓ = 13/3
Trap: a negative exponent makes a reciprocal, never a negative number (⅓, not −3). And 43/2 = (√4)³ = 8, not 6.
Worked example · Simplify fully. (3)
(16x⁸)3/4 ÷ (2x)³
✓ A: (16x⁸)3/4 = (⁴√16)³ · x⁶ = 8x⁶
✓ A: (2x)³ = 8x³
✓ CA: 8x⁶ ÷ 8x³ = x³
Worked example · Solve for x. (3)
2x3/2 = 54
✓ A: x3/2 = 27
✓ A: Raise both sides to the reciprocal exponent: x = 272/3
✓ CA: x = (∛27)² = 9
Check: 2 × 93/2 = 54. Unknowns in the exponent (2x = 32) belong to the Equations topic.
🧠 ACTIVE RECALL
Which of these numbers is irrational?
Which of these numbers is rational?
A calculator shows √3 = 1,7320508075688… How should √3 be classified?
Only one of these numbers can be written as a/b, with a and b integers and b ≠ 0. Which one?
Classify before you calculate. Which one of these is irrational?
If 10x = a, which expression equals 1000x ÷ 10x−2?
Factorise fully: 2x³ − 3x² − 8x + 12. Show every line, and name the factorising method you use on each line.
A · group in pairs: x²(2x − 3) − 4(2x − 3)Four terms means grouping. The examiner wants the same bracket in both pairs: taking out −4 turns +12 into −3.
A · take out the common bracket: (2x − 3)(x² − 4)The mark is for removing (2x − 3) as a factor of the whole expression, not of one pair.
CA · difference of two squares: (2x − 3)(x − 2)(x + 2)Fully means test every bracket. x² − 4 still breaks into (x − 2)(x + 2); stopping at (2x − 3)(x² − 4) loses this mark.
Four terms, so group in pairs and take a factor out of each pair: x²(2x − 3) − 4(2x − 3).The bracket (2x − 3) is now common to both terms, so take it out: (2x − 3)(x² − 4).x² − 4 is a difference of two squares, so the full factorisation is (2x − 3)(x − 2)(x + 2).
Simplify fully: 1/(x + 2) − 12/(x³ + 8). Write down the lowest common denominator, then show every line and say in words what each line does.
A · LCD: x³ + 8 = (x + 2)(x² − 2x + 4)The sum of cubes is the LCD. SOAP: the middle term is opposite (−2x); writing x² + 2x + 4 loses this mark and every cancel after it.
A · one fraction: numerator x² − 2x + 4 − 12 = x² − 2x − 8The first fraction is multiplied top and bottom by (x² − 2x + 4); the examiner checks that 12 is subtracted from the whole new numerator.
A · factorise the new numerator: (x − 4)(x + 2)Always try to factorise the new numerator. This is the step most answers miss, and without it nothing cancels.
CA · cancel (x + 2): (x − 4)/(x² − 2x + 4)Cancel the common factor, never terms. x² − 2x + 4 does not factorise, so the answer stops here.
x³ + 8 = (x + 2)(x² − 2x + 4), and that product is the lowest common denominator.Over the LCD the numerator is x² − 2x + 4 − 12 = x² − 2x − 8.Factorise the new numerator: x² − 2x − 8 = (x − 4)(x + 2).Cancel the common factor (x + 2) to get (x − 4)/(x² − 2x + 4).
Simplify fully, without a calculator: (6n · 4n+1) / (8n · 3n−1). Write every base as a product of primes and say in words what each line does.
A · numerator in primes: 6n = 2n · 3n and 4n+1 = 22n+2Different bases built from the same primes: 6 = 2 × 3 and 4 = 2², so (2²)ⁿ⁺¹ = 2²ⁿ⁺² because a power of a power multiplies.
A · denominator in primes: 8n = 23n8 = 2³, so 8ⁿ = 2³ⁿ; 3ⁿ⁻¹ is already a prime base and stays as it is.
A · collect prime by prime: 2² · 3¹Add the numerator's exponents and subtract the denominator's, one prime at a time. Every n cancels, which is your check that the method is right.
CA · = 12Simplify fully means a single number here; leaving 2² · 3 gives away the last mark.
Write the numerator in primes: 6n · 4n+1 = 2n · 3n · 22n+2.Write the denominator in primes: 8n · 3n−1 = 23n · 3n−1.Collect each prime: 2n+2n+2−3n · 3n−(n−1) = 2² · 3¹.So the expression simplifies to 4 × 3 = 12.
Explain to a Grade 8 learner why 3² + 3² is not 3⁴, and what it does equal.
Explain to a Grade 8 learner how to find, without a calculator, the two whole numbers that √40 lies between, and why the method works.
Explain to a Grade 8 learner why you should round only once, at the end of a calculation, and why 213 learners in 60-seater buses need 4 buses and not 3,55.
ONE-MINUTE SUMMARY Algebra opens Paper 1 and rewards precision. Squeeze a surd between perfect squares and round once, at the end. A binomial times a trinomial gives six products before you collect. Factorise in a fixed order (common factor, count the terms, test every bracket) with SOAP for the cubes. Fractions are factorise, then cancel factors only. Exponent laws act across × and ÷, never + or −: split powers, use prime bases, and read a fractional exponent as root first, power second.
IN THE EXAM
Paper 1, Question 1 in all three November sittings read: 20–22 of 100 marks, all compulsory, with no multiple choice.
Expect to place a surd, expand, factorise fully (often a cube or grouping), simplify a fraction, and evaluate exponents “without a calculator”; the commands are “factorise fully”, “simplify fully”, “determine” and “round off”.
Marks are per step (A), and consistent accuracy (CA) carries an early slip forward, so a wrong first line does not sink the question. Show every line.
Algebra and Equations together carry about 30 marks of Paper 1, and this toolkit returns in Functions and Finance.
🔒 EQUATIONS
🔒 PATTERNS
🔒 TRIGONOMETRY
🔒 FUNCTIONS
🔒 ANALYTIC GEO
🔒 EUCLID & SOLIDS
🔒
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Feeling the pressure? Take a breath.
Exam stress is completely normal. Here are a few things that actually work. None of this replaces a doctor or a trusted adult.
Breathe in
Box breathing: 4 counts in, 4 hold, 4 out, 4 hold. Do it four times.
Anxiety
Grounding: name 5 things you see, 4 you hear, 3 you can touch.
Reframe "I am going to fail" into "I have prepared, I will do my best".
Concentration
Work in 25-minute blocks, then a 5-minute break (Pomodoro).
Put the phone in another room while you study.
One task at a time. Multitasking costs more than it saves.
Sleep
Cramming all night backfires: sleep is when the brain locks in memory.
No screens for 30 minutes before bed.
The same bedtime every night keeps the brain sharp.
Study habits
Active recall (testing yourself) beats passive re-reading by far.
Space your study across days; do not cram it all into one.
Practise past papers under time. That is exactly how these notes are built.
If it gets too much, talk to a parent, teacher or a trusted adult. You do not have to carry it alone. SADAG (free, 24 hours): 0800 567 567, or SMS 31393.