Exponents, Scientific Notation and Irrational Numbers: Grade 8 Study Guide
Very large and very small numbers, and numbers that never end, have precise ways of being written.
Integer exponents follow rules: aᵐ · aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ. Scientific notation writes a number as a × 10ⁿ with 1 ≤ a < 10: the distance from Earth to the Sun, about 150,000,000 km, is 1.5 × 10⁸ km, and the width of a human hair, about 0.00007 m, is 7 × 10⁻⁵ m.
A rational number can be written as a fraction of integers; its decimal ends or repeats. An irrational number (such as √2 and π) has a decimal that never ends and never repeats. Together they form the real numbers.
Exponent laws
| Law | Example |
|---|---|
| aᵐ · aⁿ = aᵐ⁺ⁿ | 3² · 3⁴ = 3⁶ = 729 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁷ ÷ 5⁵ = 25 |
| (aᵐ)ⁿ = aᵐⁿ | (2³)² = 64 |
| a⁻ⁿ = 1/aⁿ | 10⁻² = 0.01 |
Scientific notation
To write 45,000 in scientific notation, move the decimal point to get 4.5 and count the places: 4.5 × 10⁴. To multiply, multiply the numbers and add the exponents: (3 × 10⁵)(2 × 10³) = 6 × 10⁸. To compare, look at the exponents first. Calculators show 4.5E4 for 4.5 × 10⁴.
45,000 → 4.5.
4 places.
4.5 × 10⁴.
Small numbers have negative exponents.
Irrational numbers
√4 = 2 is rational but √2 ≈ 1.414 is irrational. To estimate √20, note it lies between √16 = 4 and √25 = 5, near 4.5 (actually 4.47). Place irrational numbers on a number line by approximating. Cube roots work similarly: ∛27 = 3 and ∛10 is between 2 and 3. A repeating decimal such as 0.333… is rational (1/3); convert it by algebra.
Worked examples and common mistakes
Example: Light travels about 3 × 10⁵ km/s. In 5 seconds it goes 1.5 × 10⁶ km. Common mistakes: adding exponents when adding numbers, calling √9 irrational and mishandling negative exponents (2⁻³ is 1/8, not −8). Check by writing out the long form.
Check your understanding: model answers
1. 3.2 × 10⁻⁴. 2. 2³ = 8. 3. Irrational.
✨ Key Takeaways
- Exponent laws simplify products and quotients.
- Scientific notation: a × 10ⁿ.
- Irrational decimals never end or repeat.
- Estimate roots between perfect squares.
💬 Questions for Discussion & Reflection
- Write 0.00032 in scientific notation.
- Simplify 2⁵ · 2⁻² .
- Is √7 rational or irrational?