Ch 7 introduces cubes and cube roots. Students learn to identify perfect cubes, find cube roots using prime factorisation, and explore interesting number patterns like Hardy-Ramanujan numbers.
A perfect cube is the cube of an integer: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000… A number is a perfect cube if and only if all prime factors appear in groups of three in its prime factorisation.
Prime factorisation: factorise the number, group identical primes in triplets, take one from each group and multiply. Example: ∛5832 = ∛(2³ × 3³ × 3³) — wait, 5832 = 2³ × 3³ × 3³? Let's check: 5832 = 8 × 729 = 2³ × 9³ = (2×9)³ = 18³. So ∛5832 = 18.
1729 is the smallest number expressible as a sum of two cubes in two different ways: 1729 = 1³ + 12³ = 9³ + 10³. These are called Hardy-Ramanujan numbers or taxicab numbers.
Download: https://ncert.nic.in/textbook/pdf/hemh107.pdf | Complete book: https://ncert.nic.in/textbook/pdf/hemh1ps.zip
Find its prime factorisation. If every prime factor appears a multiple of 3 times, it's a perfect cube. Take one factor from each group of three to get the cube root.
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