Exponents and Powers: Class 7 Study Guide
An exponent tells us how many times to multiply a number by itself.
In 2⁵, the base is 2 and the exponent (power) is 5. It means 2 × 2 × 2 × 2 × 2 = 32. Any number to the power 1 is itself, and any non-zero number to the power 0 is 1. A negative exponent means a reciprocal: 2⁻³ = 1/2³ = 1/8.
The laws of exponents make sums easier.
Laws of exponents
For a non-zero number a:
| Law | Example |
|---|---|
| a^m × a^n = a^(m+n) | 2³ × 2⁴ = 2⁷ = 128 |
| a^m ÷ a^n = a^(m−n) | 5⁶ ÷ 5⁴ = 5² = 25 |
| (a^m)^n = a^(mn) | (3²)³ = 3⁶ = 729 |
| a^m × b^m = (ab)^m | 2³ × 5³ = 10³ = 1,000 |
| a⁰ = 1 | 7⁰ = 1 |
Standard form
Very large or very small numbers are written as a number between 1 and 10 times a power of 10. 3,00,000 = 3 × 10⁵. 45,000 = 4.5 × 10⁴. A small number such as 0.0007 = 7 × 10⁻⁴. The Sun is about 1.5 × 10⁸ km from the Earth.
Until one digit is before it.
That is the power.
For big numbers.
For small numbers.
Why use powers?
Powers save writing long numbers and make comparison easier. Scientists, engineers and computer experts use them every day.
Worked examples and common mistakes
Example: 2⁴ × 2² = 2⁶ = 64. Example: 6,50,00,000 = 6.5 × 10⁷. Common mistakes: multiplying the base and the exponent (2³ is 8, not 6), adding exponents when the bases differ, and treating a⁰ as 0. Any non-zero number to the power 0 is 1.
Check your understanding: model answers
1. 3² = 9. 2. 2.5 × 10⁶. 3. 1/4.
✨ Key Takeaways
- a^m × a^n = a^(m+n).
- a^m ÷ a^n = a^(m−n).
- a⁰ = 1.
- Standard form is a number from 1 to 10 times a power of 10.
💬 Questions for Discussion & Reflection
- Simplify 3⁴ ÷ 3².
- Write 25,00,000 in standard form.
- Find 2⁻² as a fraction.
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