Standard Form & Powers of 10
Standard Form & Powers of 10
Standard form (also called scientific notation) is used to write very large or very small numbers in a compact way. It is widely used in science, engineering and anywhere that extreme values appear.
What is Standard Form?
A number in standard form is written as:
A × 10ⁿ
Where:
- A is a number between 1 and 10 (1 ≤ A < 10)
- n is an integer (whole number) — positive for large numbers, negative for small numbers
Converting to Standard Form
Large numbers: Move the decimal point left until you have a number between 1 and 10. The power of 10 equals the number of places moved.
Worked Example: Write 4,560,000 in standard form.
- Move the decimal point 6 places left: 4.56
- Answer: 4.56 × 10⁶
Small numbers: Move the decimal point right. The power of 10 is negative.
Worked Example: Write 0.00037 in standard form.
- Move the decimal point 4 places right: 3.7
- Answer: 3.7 × 10⁻⁴
Converting from Standard Form
Reverse the process — move the decimal point by the power of 10.
- 2.8 × 10⁵ = 280,000
- 9.1 × 10⁻³ = 0.0091
Calculating with Standard Form
Multiplying: Multiply the A values and add the powers.
Worked Example: (3 × 10⁴) × (5 × 10⁷)
- 3 × 5 = 15
- 10⁴ × 10⁷ = 10¹¹
- 15 × 10¹¹ = 1.5 × 10¹² (adjust so A is between 1 and 10)
Dividing: Divide the A values and subtract the powers.
Worked Example: (8.4 × 10⁹) ÷ (2.1 × 10³)
- 8.4 ÷ 2.1 = 4
- 10⁹ ÷ 10³ = 10⁶
- Answer: 4 × 10⁶
Adding/Subtracting: Convert to the same power of 10 first, or convert to ordinary numbers.
Worked Example: 3.5 × 10⁴ + 6 × 10³
- 3.5 × 10⁴ = 35,000
- 6 × 10³ = 6,000
- 35,000 + 6,000 = 41,000 = 4.1 × 10⁴
Index Laws (Laws of Indices)
These rules apply to any base, not just 10:
| Rule | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁵ = 2⁸ |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁷ ÷ 5² = 5⁵ |
| (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ |
| a⁰ = 1 | 7⁰ = 1 |
| a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| a^(1/n) = ⁿ√a | 8^(1/3) = ∛8 = 2 |
| a^(m/n) = (ⁿ√a)ᵐ | 27^(2/3) = (∛27)² = 9 |
Worked Example (Higher): Simplify 16^(3/4).
- The denominator 4 means fourth root: ⁴√16 = 2
- The numerator 3 means cube: 2³ = 8
Exam Tips
- The most common mistake is writing A outside the range 1 to 10 — always adjust your answer
- On a calculator, standard form displays as e.g. 3.5E4 meaning 3.5 × 10⁴
- When adding or subtracting in standard form, converting to ordinary numbers is often the safest method