📖 Theory notes

Every note is built the same way: I can…, a quick recall, 🔑 key ideas, ✨ tips, ⚠️ common mistakes, ✏️ worked examples and 🧠 Remember in 3 lines.

🎯 Learn, then practise from two books

Each LO starts with 📘 a learning sheet (worked examples one step at a time), then 📗 Collins practice (Book 1 exercises) and, where Book 2 covers the topic, 📙 Pearson practice (Book 2 exercises, including starred questions).

🧪 LO tests · 30 minutes each

When the practice feels easy, take the timed test for that LO. Tests run in Quiz mode, submit themselves at 00:00 and can be taken once. Your report shows each LO as a pie chart.

About this chapter

Scientists, engineers and economists work with numbers like the mass of the Earth (5 970 000 000 000 000 000 000 000 kg) and the mass of an electron (0.000 … 000 91 kg, with 30 zeros). Standard form writes them short and clear: 5.97 × 10²⁴ kg and 9.1 × 10⁻³¹ kg. In this chapter you write numbers in standard form, calculate with them, and use them to solve real problems.

The chapter mixes two books. 📗 Collins Book 1 gives the method and practice for all three LOs. 📙 Pearson Book 2 adds harder calculator work (Number 7: advanced calculator problems) in LO 2 and science problems — escape velocity, gravity and the horizon — in LO 3.

LO	📗 Collins Book 1	📙 Pearson Book 2
LO 1	9.1 Standard form (Exercise 9A, 9B Q1, Q8)	—
LO 2	9.2 Calculating with standard form (Example 1, Exercise 9B)	Number 7: Example 5, Exercise 2 and 2★, Exercise 3 and 3★, Exam practice
LO 3	9.3 Solving problems (Example 2, Exercise 9C)	Number 7: Exercise 2★ Q14–15, Exercise 3 Q7–8

Learning objectives

LO 1 · 9.1 Writing numbers in standard formI can write large and small numbers in standard form a × 10ⁿ (1 ≤ a < 10), write standard-form numbers in full, and compare numbers in standard form.

LO 2 · 9.2 Calculating with standard formI can multiply, divide, add, subtract, square and root numbers in standard form, with and without a calculator.

LO 3 · 9.3 Solving problems with standard formI can solve real problems in science, space and populations using standard form, including 'how many times bigger' questions.

Check what you already know

You need	Quick check	Answer
Powers	10 × 10 × 10 × 10	10⁴ = 10 000
Multiplying by 10, 100, 1000	3.6 × 1000	3600
Dividing by 10, 100, 1000	45 ÷ 100	0.45
Index laws	10⁵ × 10³ and 10⁵ ÷ 10³	10⁸ and 10²

9.1 Writing numbers in standard form

LO 1I can… write large and small numbers in standard form a × 10ⁿ (1 ≤ a < 10), write standard-form numbers in full, and compare numbers in standard form.

🔁 Quick recall before you start
What is 10³? What is 4.5 × 100? (1000 and 450.)

📗 Collins: 9.1 Standard form, Exercise 9A (and 9B Q1, Q8)
📙 Pearson Book 2: —

Every number can be written as a number between 1 and 10 multiplied by a power of 10.

🔑 Key idea · Standard form
A number is in standard form when it is written as a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number (positive, negative or zero).

Number	Standard form	Why
625	6.25 × 10²	625 = 6.25 × 100
3147	3.147 × 10³	3147 = 3.147 × 1000
0.4	4 × 10⁻¹	0.4 = 4 × 0.1
0.000 729	7.29 × 10⁻⁴	0.000 729 = 7.29 × 0.0001

✨ Tip · Big or small?
Numbers of 10 or more have a positive power. Numbers between 0 and 1 have a negative power. The power counts how many places the decimal point moves.

⚠️ Common mistake · The first part must be between 1 and 10
52 × 10³ and 0.52 × 10⁵ are equal to 52 000 but are not in standard form. Write 5.2 × 10⁴.

✏️ Worked example · Large number

Write 46 700 in standard form.
Put the point after the first digit: 4.67.
The point moved 4 places left, so 46 700 = 4.67 × 10⁴.

✏️ Worked example · Small number

Write 0.0098 in standard form.
The first non-zero digit is 9: 9.8.
The point moved 3 places right, so 0.0098 = 9.8 × 10⁻³.

✏️ Worked example · Fixing a number that is nearly in standard form

56.7 × 10² : 56.7 = 5.67 × 10¹.
So 56.7 × 10² = 5.67 × 10¹ × 10² = 5.67 × 10³.

✨ Tip · On a calculator
Use the ×10ˣ (or EXP) key: 1.23 × 10⁻⁶ is keyed as 1 . 2 3 ×10ˣ (−) 6. Do not type × 1 0 ^ — it can upset the order of operations.

🧠 Remember in 3 lines

a × 10ⁿ with 1 ≤ a < 10.
Big number → positive power; small number → negative power.
The power = how many places the point moves.

Memory hook: One digit before the point, then count the hops.

9.2 Calculating with standard form

LO 2I can… multiply, divide, add, subtract, square and root numbers in standard form, with and without a calculator.

🔁 Quick recall before you start
Simplify 10⁴ × 10³ and 10⁴ ÷ 10⁻² using the index laws. (10⁷ and 10⁶.)

📗 Collins: 9.2 Calculating with standard form, Example 1, Exercise 9B
📙 Pearson Book 2: Number 7: Example 5, Exercise 2, 2★, 3, 3★ and Exam practice (calculator problems)

🔑 Key idea · Multiplying and dividing
Multiply (or divide) the first parts, then add (or subtract) the powers. Finally make sure the first part is between 1 and 10.

✏️ Worked example · Multiplying

2 × 10⁻² × 7 × 10⁻³ = (2 × 7) × 10⁻²⁺⁽⁻³⁾ = 14 × 10⁻⁵
14 is too big: 14 × 10⁻⁵ = 1.4 × 10¹ × 10⁻⁵ = 1.4 × 10⁻⁴.

✏️ Worked example · Dividing

(1.2 × 10⁴) ÷ (6 × 10⁴) = (1.2 ÷ 6) × 10⁴⁻⁴ = 0.2 × 10⁰
0.2 is too small: 0.2 = 2 × 10⁻¹, so the answer is 2 × 10⁻¹.

🔑 Key idea · Adding and subtracting
You cannot add or subtract the powers. Write both numbers with the same power (or as ordinary numbers), then add or subtract the first parts.

✏️ Worked example · Adding

4.8 × 10⁶ + 1.6 × 10⁴
1.6 × 10⁴ = 0.016 × 10⁶
(4.8 + 0.016) × 10⁶ = 4.816 × 10⁶

🔑 Key idea · Powers and roots
(a × 10ⁿ)² = a² × 10²ⁿ. √(a × 10²ⁿ) = √a × 10ⁿ. For example √(36 × 10⁻⁴) = 6 × 10⁻².

✨ Tip · Calculator questions (Pearson Number 7)
Type the whole calculation in one go, using brackets around every numerator, denominator and root, and the ×10ˣ key. Only round at the very end, then give the answer to the accuracy asked (often 3 significant figures).

✏️ Worked example · Pearson Example 5a

(1.76 × 10³ ÷ √(5.3 × 10⁻²))⁴
Keys: ( 1.76 ×10ˣ 3 ÷ √( 5.3 ×10ˣ −2 ) ) xⁿ 4 =
3.41585… × 10¹⁵ = 3.42 × 10¹⁵ (3 s.f.)

⚠️ Common mistake · Adding the powers when adding numbers
3 × 10⁴ + 2 × 10³ is not 5 × 10⁷. Write 3 × 10⁴ + 0.2 × 10⁴ = 3.2 × 10⁴.

🧠 Remember in 3 lines

× and ÷: work with the numbers, then add or subtract the powers.
+ and −: make the powers the same first.
Always finish with 1 ≤ a < 10.

Memory hook: Times → add powers. Plus → same powers.

9.3 Solving problems with standard form

LO 3I can… solve real problems in science, space and populations using standard form, including 'how many times bigger' questions.

🔁 Quick recall before you start
Speed = distance ÷ time. If a car travels 120 km at 60 km/h, how long does it take? (2 hours.)

📗 Collins: 9.3 Solving problems, Example 2, Exercise 9C
📙 Pearson Book 2: Number 7: Exercise 2★ Q14–15 (escape velocity, gravity), Exercise 3 Q7–8 (light from Mars, distance to the horizon)

Real problems give you the numbers in words or tables. Turn them into standard form, choose the operation, then calculate.

Words	Standard form
1 thousand	10³
1 million	10⁶
1 billion (1000 million)	10⁹
1 km	10³ m
1 tonne	10³ kg = 10⁶ g

✏️ Worked example · Light from the Sun (Collins Example 2)

Distance 150 million km = 1.5 × 10⁸ km. Speed 3 × 10⁵ km/s.
Time = distance ÷ speed = 1.5 × 10⁸ ÷ 3 × 10⁵ = 0.5 × 10³ = 500 s
500 s = 8 minutes 20 seconds.

🔑 Key idea · How many times bigger?
Divide the bigger number by the smaller. Saturn ÷ Earth = 5.686 × 10²⁶ ÷ 6.04 × 10²¹ = 9.41 × 10⁴, so Saturn is about 94 000 times heavier.

✏️ Worked example · Population density

Density = population ÷ area.
Sri Lanka: 1.91 × 10⁷ ÷ 6.56 × 10⁴ = 291 people per km².
Russian Federation: 1.43 × 10⁸ ÷ 1.71 × 10⁷ = 8.4 people per km².

⚠️ Common mistake · Mixing units
Change everything to the same units first. 57.6 million km = 5.76 × 10⁷ km = 5.76 × 10¹⁰ m. Light from Mars then takes 5.76 × 10¹⁰ ÷ 3 × 10⁸ = 192 s.

✨ Tip · Sense check
Ask 'Is my answer sensible?' Light reaching the Earth from the Sun in 500 s (about 8 minutes) is right; 5 s or 50 000 s would not be.

🧠 Remember in 3 lines

Turn words into standard form (million = 10⁶).
Same units, then choose ×, ÷, + or −.
'How many times bigger' means divide.

Memory hook: Words → powers → operation → check.