Standard Form – Grade 9 IGCSE Maths

Welcome to Standard Form!

Standard form (also called scientific notation) is a way of writing very large or very small numbers concisely. Scientists use it to express distances in space, sizes of atoms, and populations. It is an essential skill for IGCSE Mathematics and Science.

a × 10ⁿ where 1 ≤ a < 10 and n is an integer

What is Standard Form?

The a × 10ⁿ format

Large Numbers

Positive powers of 10

Small Numbers

Negative powers of 10

Multiplying

Multiply coefficients, add powers

Dividing

Divide coefficients, subtract powers

Adding & Subtracting

Convert to same power first

1. What is Standard Form?

Standard form writes any number as a × 10ⁿ where a is between 1 and 10 (not including 10) and n is an integer (can be positive, negative, or zero).

a × 10ⁿ where 1 ≤ a < 10, n ∈ ℤ

Valid standard form: 3.2 × 10⁴, 1.0 × 10⁻³, 9.99 × 10⁸
NOT valid: 12 × 10³ (12 ≥ 10), 0.5 × 10⁴ (0.5 < 1), 3.2 × 5² (must be power of 10)

The rule 1 ≤ a < 10 means a can equal 1 but must be strictly less than 10. So 1.0 × 10⁵ is fine, but 10 × 10⁵ is not — it should be 1.0 × 10⁶.

2. Writing Large Numbers in Standard Form

Count how many places you move the decimal point to the LEFT to get a number between 1 and 10. That count becomes your positive power.

4,500 → standard form:
Move decimal 3 places left: 4.500 → 4.5
So 4,500 = 4.5 × 10³

85,000,000 → standard form:
Move decimal 7 places left: 8.5000000
So 85,000,000 = 8.5 × 10⁷

3. Writing Small Numbers in Standard Form

For numbers less than 1, move the decimal point to the RIGHT to get a number between 1 and 10. The power becomes negative.

0.0056 → standard form:
Move decimal 3 places right: 5.6
So 0.0056 = 5.6 × 10⁻³

0.000045 → standard form:
Move decimal 5 places right: 4.5
So 0.000045 = 4.5 × 10⁻⁵

Memory aid: BIG numbers → positive power (decimal goes LEFT to get a). SMALL numbers → negative power (decimal goes RIGHT).

4. Multiplying in Standard Form

(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10^(m+n)

Example: (3 × 10⁴) × (4 × 10⁵)
Multiply coefficients: 3 × 4 = 12
Add powers: 4 + 5 = 9
Result: 12 × 10⁹ — but 12 ≥ 10, so adjust:
= 1.2 × 10¹⁰

Always check after multiplying whether the coefficient is still between 1 and 10. If not, adjust: e.g. 12 × 10⁹ = 1.2 × 10¹⁰.

5. Dividing in Standard Form

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10^(m−n)

Example: (8 × 10⁸) ÷ (4 × 10⁵)
Divide coefficients: 8 ÷ 4 = 2
Subtract powers: 8 − 5 = 3
Result: 2 × 10³

6. Adding and Subtracting in Standard Form

You cannot directly add numbers in standard form unless they have the same power of 10. Convert both to the same power (or to ordinary numbers), then add.

Example: 3 × 10⁴ + 2 × 10³
Convert to same power: 3 × 10⁴ = 30 × 10³
30 × 10³ + 2 × 10³ = 32 × 10³ = 3.2 × 10⁴

For addition/subtraction, it's often easiest to convert both numbers to ordinary form, add them, then convert back to standard form.

Example 1 — Write 320,000 in standard form

Count decimal moves left: 3.20000 — moved 5 places

Answer: 3.2 × 10⁵

Example 2 — Write 0.000082 in standard form

Count decimal moves right: 8.2 — moved 5 places right

Answer: 8.2 × 10⁻⁵

Example 3 — Multiply: (2 × 10³) × (3 × 10⁴)

Coefficients: 2 × 3 = 6

Powers: 3 + 4 = 7

Answer: 6 × 10⁷

Example 4 — Divide: (9 × 10⁶) ÷ (3 × 10²)

Coefficients: 9 ÷ 3 = 3

Powers: 6 − 2 = 4

Answer: 3 × 10⁴

Example 5 — Convert to ordinary number: 7.2 × 10⁻³

Negative power means small number. Move decimal 3 places LEFT from 7.2.

7.2 → 0.0072. Answer: 0.0072

Example 6 — Add: 4 × 10⁵ + 3 × 10⁴

Convert: 4 × 10⁵ = 400,000 3 × 10⁴ = 30,000

Sum = 430,000 = 4.3 × 10⁵

Number → Standard Form Converter

Enter any positive number to see it converted to standard form with full working.

Number:

Result will appear here.

Standard Form → Ordinary Number

Enter a and n to convert a × 10ⁿ to an ordinary number.

a = n =

Result will appear here.

Exercise 1 — Finding the Power

Write each number in standard form a × 10ⁿ. Enter the value of n (the power of 10).

1. Write 4500 in standard form. Give n (4.5 × 10ⁿ).

2. Write 320,000 in standard form. Give n.

3. Write 85,000,000 in standard form. Give n.

4. Write 0.0056 in standard form. Give n (negative).

5. Write 0.000045 in standard form. Give n (negative).

6. Write 260 in standard form. Give n.

7. Write 7,200,000 in standard form. Give n.

8. Write 0.00082 in standard form. Give n (negative).

9. Write 59,000 in standard form. Give n.

10. Write 0.73 in standard form. Give n.

Exercise 2 — Standard Form to Ordinary Number

Convert each standard form number to an ordinary number. Type the full number.

1. 4.5 × 10³ = ?

2. 3.2 × 10⁵ = ?

3. 8.5 × 10⁷ = ?

4. 5.6 × 10⁻³ = ?

5. 4.5 × 10⁻⁵ = ?

6. 2.6 × 10² = ?

7. 7.2 × 10⁶ = ?

8. 8.2 × 10⁻⁴ = ?

9. 5.9 × 10⁴ = ?

10. 7.3 × 10⁻¹ = ?

Exercise 3 — Multiplying in Standard Form (Coefficient)

Multiply the numbers in standard form. Give the coefficient a (to 2 d.p.) in the answer a × 10ⁿ.

1. (3 × 10²) × (4 × 10⁵) = a × 10⁷. Find a.

2. (4 × 10³) × (6 × 10⁵) = a × 10⁹. Find a.

3. (6 × 10⁴) × (6 × 10⁵) = a × 10¹⁰. Find a.

4. (2 × 10²) × (8 × 10²) = a × 10⁵. Find a.

5. (8 × 10³) × (6 × 10⁴) = a × 10⁸. Find a.

6. (5 × 10²) × (3 × 10³) = a × 10⁶. Find a.

7. (7 × 10³) × (3 × 10⁶) = a × 10¹⁰. Find a.

8. (4 × 10¹) × (8 × 10²) = a × 10⁴. Find a.

9. (9 × 10⁵) × (2 × 10⁶) = a × 10¹². Find a.

10. (8 × 10¹) × (8 × 10¹) = a × 10³. Find a.

Exercise 4 — Multiplying in Standard Form (Power)

Multiply and give the power n in the final answer a × 10ⁿ (after adjusting so 1 ≤ a < 10).

1. (3 × 10⁴) × (4 × 10²). Final answer: a × 10ⁿ. Give n.

2. (5 × 10⁵) × (6 × 10³). Give n.

3. (3 × 10⁶) × (4 × 10⁴). Give n.

4. (2 × 10³) × (4 × 10¹). Give n.

5. (4 × 10⁴) × (6 × 10³). Give n.

6. (3 × 10³) × (5 × 10²). Give n.

7. (5 × 10⁵) × (4 × 10⁴). Give n.

8. (2 × 10²) × (4 × 10¹). Give n.

9. (3 × 10⁶) × (4 × 10⁵). Give n.

10. (1.5 × 10²) × (2 × 10⁰). Give n in 3 × 10ⁿ.

Exercise 5 — Dividing in Standard Form

Divide. Give the coefficient a (to 2 d.p.) in the result a × 10ⁿ.

1. (8 × 10⁶) ÷ (4 × 10³). Result = a × 10³. Give a.

2. (9 × 10⁷) ÷ (3 × 10⁴). Result = a × 10³. Give a.

3. (6 × 10⁵) ÷ (4 × 10³). Result = a × 10². Give a.

4. (8 × 10⁸) ÷ (2 × 10⁵). Result = a × 10³. Give a.

5. (5 × 10⁶) ÷ (2 × 10⁵). Result = a × 10¹. Give a.

6. (6 × 10⁴) ÷ (5 × 10³). Result = a × 10¹. Give a.

7. (7 × 10⁵) ÷ (2 × 10³). Result = a × 10². Give a.

8. (1.2 × 10⁶) ÷ (5 × 10⁴). Result = a × 10². Give a.

9. (1 × 10⁷) ÷ (2 × 10⁵). Result = a × 10². Give a.

10. (8 × 10⁵) ÷ (5 × 10⁴). Result = a × 10¹. Give a.

Practice — 20 Questions

Mixed practice on all standard form skills.

1. Write 4500 in standard form. Give the power n.

2. Write 320,000 in standard form. Give n.

3. Write 85,000,000 in standard form. Give n.

4. Write 0.0056 in standard form. Give n.

5. Write 0.000045 in standard form. Give n.

6. Write 260 in standard form. Give n.

7. Write 7,200,000 in standard form. Give n.

8. Write 0.00082 in standard form. Give n.

9. Write 59,000 in standard form. Give n.

10. Write 0.73 in standard form. Give n.

11. 4.5 × 10³ as ordinary number?

12. 3.2 × 10⁵ as ordinary number?

13. 5.6 × 10⁻³ as ordinary number?

14. 4.5 × 10⁻⁵ as ordinary number?

15. 2.6 × 10² as ordinary number?

16. 7.2 × 10⁶ as ordinary number?

17. (3 × 10²) × (4 × 10⁵). Give coefficient a.

18. (4 × 10³) × (6 × 10⁵). Give coefficient a.

19. (6 × 10⁴) × (6 × 10⁵). Give coefficient a.

20. (2 × 10²) × (8 × 10²). Give coefficient a.

Challenge — 8 Questions

Multiplication in standard form. Give the coefficient a (to 2 d.p.) in a × 10ⁿ.

1. (3 × 10²) × (4 × 10⁵). Give coefficient a.

2. (4 × 10³) × (6 × 10⁵). Give coefficient a.

3. (6 × 10⁴) × (6 × 10⁵). Give coefficient a.

4. (2 × 10²) × (8 × 10²). Give coefficient a.

5. (8 × 10³) × (6 × 10⁴). Give coefficient a.

6. (5 × 10²) × (3 × 10³). Give coefficient a.

7. (7 × 10³) × (3 × 10⁶). Give coefficient a.

8. (4 × 10¹) × (8 × 10²). Give coefficient a.