Cubes and Cube Roots

3

Step 1: Find prime factors of 1029. Step 2: 1029 ÷ 3 = 343; 343 = 7 × 7 × 7. Step 3: So 1029 = 3 × 7 × 7 × 7 = 3 × (7×7×7). Step 4: The triplet of 7 is complete, but 3 appears only once (incomplete triplet). Step 5: Divide by 3 → 1029 ÷ 3 = 343 = 7³, which is a perfect cube. The answer is 3.

0.6

Step 1: Convert 0.216 to a fraction: 0.216 = 216/1000. Step 2: Find cube root of numerator: ∛216 = ∛(6×6×6) = 6. Step 3: Find cube root of denominator: ∛1000 = ∛(10×10×10) = 10. Step 4: So ∛(216/1000) = 6/10 = 0.6. Step 5: Verify: 0.6³ = 0.6×0.6×0.6 = 0.216. ✓ A common mistake is to forget to convert to a fraction first.

-729

Step 1: We need (-9)³ = (-9) × (-9) × (-9). Step 2: (-9) × (-9) = +81 (negative × negative = positive). Step 3: 81 × (-9) = -729 (positive × negative = negative). Step 4: So (-9)³ = -729. Remember: the cube of a negative number is always negative because you multiply negative sign an odd number (3) of times. A common mistake is to write +729 by forgetting the sign.

-8

Step 1: ∛(-512) = -(∛512), since cube root of a negative number is negative. Step 2: Find ∛512: prime factorise 512 = 2⁹ = (2×2×2) × (2×2×2) × (2×2×2). Step 3: ∛512 = 2 × 2 × 2 = 8. Step 4: Therefore ∛(-512) = -8. Step 5: Verify: (-8)³ = (-8)×(-8)×(-8) = 64×(-8) = -512. ✓