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PERPANGKATAN DUA DAN TIGA SUATU BILANGAN

PERPANGKATAN DUA DAN TIGA SUATU BILANGAN. Pangkat Dua Suatu Bilangan. 4 2 = 4 × 4 = 16. = (4 – 1) × (4 + 1) + 1 2 = 3 × 5 + 1 = 16 = (4 – 2) × (4 + 2) + 2 2 = 2 × 6 + 4 = 16 = (4 – 3) × (4 + 3) + 3 2 = 1 × 7 + 9 = 16. 8 2 = 8 × 8 = 64. = (8 – 1) × (8 + 1) + 1 2 = 7 × 9 + 1 = 64

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PERPANGKATAN DUA DAN TIGA SUATU BILANGAN

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  1. PERPANGKATAN DUA DAN TIGA SUATU BILANGAN

  2. Pangkat Dua Suatu Bilangan 42 = 4 × 4 = 16 = (4 – 1) × (4 + 1) + 12 = 3 × 5 + 1 = 16 = (4 – 2) × (4 + 2) + 22 = 2 × 6 + 4 = 16 = (4 – 3) × (4 + 3) + 32 = 1 × 7 + 9 = 16 82 = 8 × 8 = 64 = (8 – 1) × (8 + 1) + 12 = 7 × 9 + 1 = 64 = (8 – 2) × (8 + 2) + 22 = 6 × 10 + 4 = 64 = (8 – 3) × (8 + 3) + 32 = 5 × 11 + 9 = 64 = (8 – 4) × (8 + 4) + 42 = 5 × 12 + 16 = 64

  3. 122 = 12 × 12 = 144 = (12 – 1) × (12+ 1) + 12 = 11 × 13 + 1 = 144 = (12 – 2) × (12 + 2) + 22 = 10 × 14 + 4 = 144 = (12 – 3) × (12 + 3) + 32 = 9 × 15 + 9 = 144 = (12 – 4) × (12 + 4) + 42 = 8 × 16 + 16 = 144 962 = 96 × 96 = 9216 = (96 – 4) × (96 + 4) + 42 = 92 × 100 + 16 = 9216 Secara umum diperoleh : a2 = (a – b) × (a + b) + a2

  4. 9992 = 999 × 999 = 998.001 = (103 – 3) × (103 + 3) + 32 = 100 × 106 + 9 = 10.609 1032 = 103 × 103 = 10.609 = 100 × 106 + 9 = 10.609

  5. Penarikan Akar Pangkat Dua • Teknik Perkiraan • Faktorisasi prima • Teknik Calandra

  6. Teknik Perkiraan • Fakta Dasar Penarikan Akar Pangkat Dua 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 • Puluhannya 20 → 202 = 400 30 → 302 = 900 • Gunakan 20 yang ≤ 529 • Satuannya → Nilai satuan angka 9 adalah 3 dan 7 • Dicoba yang hasilnya benar 232 = 529 272 = 729 • Jadi 232 = 529

  7. Faktorisasi Prima

  8. Penarikan Akar Pangkat Tiga • Teknik Perkiraan • Faktorisasi Prima • Teknik Calandra

  9. 1. Teknik Perkiraan 13 = 1 23 = 8 33 = 27 43 = 64 53 = 125 63 = 216 73 = 343 83 = 512 93 = 729 103 = 1000 1 – 1 2 – 8 3 – 7 4 – 4 5 – 5 6 – 6 7 – 3 8 – 2 9 – 9 10 - 0 Harus hafal bilangan kubik di ≤1000

  10. 13 = 1(mendekati 4) 6 16

  11. 2 3 = 8 (mendekati 19) 7 27

  12. 2. Faktorisasi Prima 1000 2 500 2 8 250 2 2 125 4 5 2 25 5 2 5

  13. 3. Teknik Calandra • Kelompokkan tiga angka dari belakang • Cari bilangan terbesar yang jika dipangkatkan tiga hasilnya ≤12. bilangan itu adalah 2 sebab 23 ≤ 12 = 2 23 = 8 4 812

  14. 2 artinya 2 puluhan atau 20 • Cari bilangan terbesar yang memenuhi hubungan : [(3 × 202) + 3 × 20 × … + …2] × …≤ 4812 Bilangan tersebut adalah 3, sebab (1200 + 60 × 3 + 32] × 3≤ 4812 (1200 + 60 × 3 + 32) × 3 = 4 617 = 23 23 = 8 4 812 645 904

  15. 23 artinya 23 puluhan (230). • Cari bilangan terbesar yang memenuhi hubungan : • [(3  2302) + (3  230)  … + …2]  …≤ 195904 • Bilangan itu adalah 4, karena : • (158700 + 690  4 + 42)  4 ≤ 195904

  16. 22 = 8 [(3 × 202) + 3 × 20 × … + …2] × … ≤ 4812 (1200 + 60 × 3 + 32) × 3 = 4617 [(3  2302) + (3  230)  … + …2]  … ≤ 195904 (158700 + 690  4 + 42) 4 = 195904 0 Teknik calandra secara singkat

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