| --- |
| source: ncert |
| topic: numbers |
| class_level: class_8 |
| chapter: ch12 |
| difficulty: intermediate |
| --- |
| |
| Class 8 | Ch12: Exponents and Powers |
| NEGATIVE EXPONENTS: |
| a^(-n)=1/a^n for any non-zero a. 2^(-3)=1/8. 10^(-4)=0.0001. |
| a^(-1)=1/a (reciprocal). |
| LAWS (extend from Class 7 to include negative exponents): |
| a^m × a^n=a^(m+n). a^m ÷ a^n=a^(m-n). (a^m)^n=a^(mn). |
| a^m × b^m=(ab)^m. a^m ÷ b^m=(a/b)^m. a^0=1. |
| These ALL work for negative exponents too. |
| STANDARD FORM (SCIENTIFIC NOTATION): |
| Number between 1 and 10 multiplied by integer power of 10. |
| Large: 6,020,000,000=6.02×10^9. |
| Small: 0.000000000016=1.6×10^(-11). |
| To compare: larger power of 10=larger number. |
| EXPRESSING IN STANDARD FORM: |
| Move decimal point until you get a number between 1 and 10. |
| Number of places moved = power of 10 (positive if moved left, negative if moved right). |
| 485 → move 2 left → 4.85×10^2. |
| 0.00605 → move 3 right → 6.05×10^(-3). |
| SOLVED EXAMPLES: |
| Example 1: (2^(-3)×3^(-2))÷(6^(-1)). |
| =1/8×1/9÷1/6=1/72×6=6/72=1/12. |
| Example 2: (5/3)^(-2)×(3/5)^3=(3/5)^2×(3/5)^3=(3/5)^5=243/3125. |
| Example 3: Express 0.00000605 in standard form. |
| Move decimal 6 places right → 6.05×10^(-6). |
| COMMON MISTAKES: |
| 2^(-3) = 1/8 NOT -8. Negative exponent means reciprocal, not negative number. |
| (a/b)^(-n)=(b/a)^n. So (2/3)^(-2)=(3/2)^2=9/4. |
| In scientific notation, first number MUST be between 1 and 10. |
