Types of Number, Powers, Roots and Ordering at Extended Level
Primes, HCF/LCM, rational vs irrational, roots, ordering
Quick recap of the families. Natural numbers are 1, 2, 3, … Integers add 0 and the negatives. A prime has exactly two factors (so 1 is NOT prime). You should also recognise square numbers (1, 4, 9, 16, …), cube numbers (1, 8, 27, 64, …) and triangle numbers (1, 3, 6, 10, 15, …).
Prime factorisation writes a number as a product of primes in index form: \(360 = 2^{3} \times 3^{2} \times 5\). At Extended level you use it to find the HCF and LCM of two or three numbers at once:
Rational or irrational? A number is rational if it can be written as \(\dfrac{a}{b}\) with \(a\) and \(b\) integers (\(b \neq 0\)). That includes integers, terminating decimals and recurring decimals — for example \(0.\dot{2}\dot{7} = \dfrac{27}{99} = \dfrac{3}{11}\). A number is irrational if its decimal goes on forever with no repeating pattern: \(\pi\), and surds such as \(\sqrt{2}\), \(\sqrt{12}\) and \(\sqrt[3]{10}\). Careful: \(\sqrt{2.25} = 1.5\) and \(\sqrt{\dfrac{9}{4}} = \dfrac{3}{2}\) look like surds but are rational.
Reciprocals. The reciprocal of \(x\) is \(\dfrac{1}{x}\). Turn a mixed number into an improper fraction first, then flip: reciprocal of \(1\dfrac{3}{5} = \dfrac{8}{5}\) is \(\dfrac{5}{8}\). For a decimal, write it as a fraction: \(0.4 = \dfrac{2}{5}\), so its reciprocal is \(\dfrac{5}{2} = 2.5\).
Powers and roots of fractions and decimals. Apply the power or root to the top AND the bottom: \(\left(\dfrac{2}{3}\right)^{3} = \dfrac{8}{27}\), \(\sqrt{0.0081} = 0.09\) (because \(0.09^{2} = 0.0081\)), \(\sqrt[3]{-0.125} = -0.5\). Cube roots of negatives exist; square roots of negatives do not (in IGCSE).
Ordering. To order a mixed list (fractions, decimals, percentages, roots, standard form, negatives), change everything to decimals with the same number of places, then compare. Use the symbols \(=\), \(\neq\), \(<\), \(>\), \(\leq\), \(\geq\) correctly: \(x \leq 2\) includes 2; \(x < 2\) does not.
Find the HCF and the LCM of 84, 120 and 180.
\(84 = 2^{2} \times 3 \times 7\) \(120 = 2^{3} \times 3 \times 5\) \(180 = 2^{2} \times 3^{2} \times 5\)
HCF: only 2 and 3 are in all three. Lowest powers: \(2^{2} \times 3 = 12\).
LCM: every prime (2, 3, 5, 7), highest powers: \(2^{3} \times 3^{2} \times 5 \times 7 = 2520\).
Three combis leave the Gaborone bus rank together at 06:00. Route A leaves every 12 minutes, route B every 18 minutes and route C every 30 minutes. When do all three next leave together?
“Next time together” means the LCM.
\(12 = 2^{2} \times 3\), \(18 = 2 \times 3^{2}\), \(30 = 2 \times 3 \times 5\) ⇒ LCM \(= 2^{2} \times 3^{2} \times 5 = 180\) minutes
180 minutes = 3 hours, so they next leave together at 09:00.
Without a calculator: (a) write down the reciprocal of \(1\dfrac{3}{5}\) and of 0.4; (b) evaluate \(\left(\dfrac{2}{3}\right)^{3}\), \(\sqrt{0.0081}\) and \(\left(1\dfrac{1}{2}\right)^{2}\).
(a) \(1\dfrac{3}{5} = \dfrac{8}{5}\) → reciprocal \(\dfrac{5}{8}\). \(0.4 = \dfrac{2}{5}\) → reciprocal \(\dfrac{5}{2} = 2.5\).
(b) \(\left(\dfrac{2}{3}\right)^{3} = \dfrac{2^{3}}{3^{3}} = \dfrac{8}{27}\). \(\sqrt{0.0081} = 0.09\). \(\left(1\dfrac{1}{2}\right)^{2} = \left(\dfrac{3}{2}\right)^{2} = \dfrac{9}{4} = 2\dfrac{1}{4}\) (not \(1\dfrac{1}{4}\)!).
Write these in order, smallest first: \(\dfrac{5}{8}\), \(0.\dot{6}\), 62%, \(\sqrt{0.4}\), 0.6
Convert to 4 decimal places: \(\dfrac{5}{8} = 0.6250\), \(0.\dot{6} = 0.6667\), 62% = 0.6200, \(\sqrt{0.4} = 0.6325\) (it lies between 0.63 and 0.64, since \(0.63^{2} = 0.3969\) and \(0.64^{2} = 0.4096\)), 0.6 = 0.6000.
\(0.6 < 62\% < \dfrac{5}{8} < \sqrt{0.4} < 0.\dot{6}\)
Show that \(0.\dot{2}\dot{7}\) is rational, and say whether \(\sqrt{12}\) is rational or irrational.
Let \(x = 0.272727\ldots\). Then \(100x = 27.2727\ldots\). Subtract: \(99x = 27\), so \(x = \dfrac{27}{99} = \dfrac{3}{11}\). It is a fraction of integers, so it is rational.
\(\sqrt{12} = 2\sqrt{3}\). 12 is not a square number, so \(\sqrt{12}\) is a surd: irrational.
Q1. Write 360 as a product of its prime factors. Give your answer in index form.
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= 23 × 32 × 5 ✓ A1
Q2. Find (a) the highest common factor (HCF) and (b) the lowest common multiple (LCM) of 72, 108 and 180.
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(a) HCF = 22 × 32 = 36 ✓ A1
(b) LCM = 23 × 33 × 5 = 1080 ✓ A1
Q3. \(A = 2^{4} \times 3 \times 5^{2}\) and \(B = 2^{2} \times 3^{3} \times 7\). Find the HCF of \(A\) and \(B\), and write the LCM of \(A\) and \(B\) in index form.
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LCM = 24 × 33 × 52 × 7 ✓ B1
Q4. A rectangular floor in a Maun lodge measures 3.6 m by 4.8 m. It is covered with identical square tiles, with no cutting. Find the side length of the largest tile that can be used, and the number of these tiles needed.
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HCF = 23 × 3 × 5 = 120 cm ✓ A1
360 ÷ 120 = 3, 480 ÷ 120 = 4, 3 × 4 = 12 tiles ✓ A1
Q5. From this list, write down all the irrational numbers.
\(\sqrt{16}\) \(\sqrt{20}\) \(0.\dot{4}\) \(\pi + 1\) \(\dfrac{3}{7}\) \(\sqrt{2.25}\) \(\sqrt[3]{10}\)
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(B1 for two correct and none wrong, or three correct and one wrong)
Q6. Write down the reciprocal of (a) \(2\dfrac{3}{4}\) (b) 0.08
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(b) 0.08 = 8100 → 1008 = 12.5 ✓ B1
Q7. Evaluate (a) \(\sqrt[3]{-\dfrac{64}{125}}\) (b) \(0.3^{3}\) (c) \(\sqrt{1\dfrac{9}{16}}\)
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(b) 0.3 × 0.3 × 0.3 = 0.027 ✓ B1
(c) √2516 = 54 ✓ B1
Q8. Work out \(\sqrt[4]{1296} + 2.5^{3}\).
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6 + 15.625 = 21.625 ✓ A1
Q9. Write these numbers in order of size, starting with the smallest.
\(3.1 \times 10^{-2}\) 0.0302 \(\dfrac{1}{32}\) 0.3%
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0.3%, 0.0302, 3.1 × 10−2, 132 ✓ A1
Q10. Write one of the symbols \(=\), \(<\) or \(>\) in each box.
(a) \(\sqrt{50}\ \square\ 7\) (b) \((-2)^{3}\ \square\ -2^{3}\) (c) \(\dfrac{3}{11}\ \square\ 0.27\)
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(b) −8 and −8 so (−2)3 = −23 ✓ B1
(c) 311 = 0.272727… > 0.27 ✓ B1
Q11. \(n\) is an integer and \(-3 < n \leq 2\). List all the possible values of \(n\). Which of them are prime?
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prime: 2 only ✓ B1
- Mixing up HCF and LCM powers: HCF takes the LOWEST power of the shared primes; LCM takes the HIGHEST power of every prime.
- Calling recurring decimals irrational: \(0.\dot{3} = \dfrac{1}{3}\) is rational. Only non-repeating, never-ending decimals (π, surds) are irrational.
- Flipping a mixed number without converting: the reciprocal of \(2\dfrac{3}{4}\) is \(\dfrac{4}{11}\), not \(2\dfrac{4}{3}\).
- Splitting a root over addition: \(\sqrt{1\dfrac{9}{16}} \neq 1\dfrac{3}{4}\). Make one fraction first: \(\sqrt{\dfrac{25}{16}} = \dfrac{5}{4}\).
- Ordering negatives the wrong way: −0.5 is SMALLER than −0.35. On a number line, further left is smaller.
- Ignoring the end of an inequality: \(<\) leaves the end value out; \(\leq\) puts it in.
- Paper 2 (no calculator): “Write 360 as a product of its prime factors” [2] and “Find the HCF / LCM of …” [2–3] appear very often, sometimes with numbers given already in index form (Q3 style).
- Word problems hide HCF/LCM: “largest tile”, “next time together”, “smallest number of packs”. Decide first whether you need the HCF (dividing up) or the LCM (meeting again).
- Command words: “Write down” (1 mark, no working needed), “Find” or “Work out” (show steps), “Show that” (every step must be written).
- Ordering questions are usually [2]: one mark for correct decimal conversions, one for the correct order in the original form.
- Rational/irrational and reciprocal questions are short [1–2] items, but they feed directly into the surds unit (N-E6) on Paper 4.