| --- |
| source: ncert |
| topic: numbers |
| class_level: class_8 |
| chapter: ch7 |
| difficulty: beginner |
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| |
| 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. |
