source: ncert
topic: numbers
class_level: class_8
chapter: ch7
difficulty: beginner
Class 8 | Ch7: Cubes and Cube Roots PERFECT CUBES: 1,8,27,64,125,216,343,512,729,1000... 1^3=1, 2^3=8, 3^3=27, 4^3=64, 5^3=125, 6^3=216, 7^3=343, 8^3=512, 9^3=729, 10^3=1000. PROPERTIES OF CUBES: Cube of even number is even. Cube of odd number is odd. Cube of negative number is negative: (-2)^3=-8. A perfect cube can end in any digit 0-9 (unlike squares). CUBE ROOT BY PRIME FACTORISATION: Group prime factors in sets of 3. Take one from each group. cbrt(13824): 13824=2^9×3^3=(2^3)^3×3^3=(8×3)^3=24^3. So cbrt(13824)=24. FINDING SMALLEST MULTIPLIER/DIVISOR: To make a perfect cube: each prime factor must appear in multiples of 3. 675=3^3×5^2. Need one more 5^1 to make 5^3. Multiply by 5. 675×5=3375=15^3. SOLVED EXAMPLES: Example 1: cbrt(13824). 13824=2×6912=2×2×3456=...=2^9×3^3. cbrt=2^3×3=24. Example 2: Is 1728 a perfect cube? 1728=2^6×3^3=(2^2×3)^3=12^3. YES, cbrt=12. Example 3: Find smallest number to divide 81 to make perfect cube. 81=3^4. Need to remove one 3 (keep 3^3). Divide by 3. 81÷3=27=3^3. Answer: 3. Example 4: What is cube of -5? (-5)^3=-125. COMMON MISTAKES: cbrt(-8)=-2 (negative cube root exists, unlike square roots). cbrt(a^3+b^3) ≠ a+b. Must factorise properly. Perfect cube can end in any digit, not restricted like perfect squares.