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.