Pada potensi kali ini kita akan memakai sifat perkalian istimewa bentuk aljabar untuk menjumlah perkalian bilangan bulat biasa, sehingga perkalian dapat teratasi dgn lebih cepat. Ada beberapa bentuk perkalian istimewa untuk menghitung perkalian bilangan yg dapat kita pelajar, ialah sebagai berikut.
1. Bentuk a(b + c) = ab + ac
Untuk memahami bentuk ini, amati pola berikut.
Contoh:
1. 8 × 46 = 8 × (40 + 6)
= 8 × 40) + (8 × 6)
= 320 + 48
= 368
2. 9 × 78 = 9 × (70 + 8)
= (9 × 70) + (9 × 8)
= 630 + 72
= 702
Cara lain:
1. 8 × 46 = 8 × (50 – 4)
= (8 × 50) – (8 × 4)
= 400 – 32
= 368
2. 9 × 78 = 9 × (80 – 2)
= (9 × 80) – (9 × 2)
= 720 – 18
= 702
2. Bentuk (x + p)(x + q) = x2 + (p + q)x + pq
Untuk memahami bentuk ini, perhatikan teladan berikut.
Contoh:
1. 77 × 73 = (70 + 7)(70 + 3)
= (70)2 + (7 + 3)70 + (7)(3)
= 4.900 + (10)70 + 21
= 4.900 + 700 + 21
= 5.621
2. 93 × 84 = (90 + 3)(90 – 6)
= (90)2 + (3 – 6)90 + (3)(-6)
= 8.100 + (-3)90 – 18
= 8.100 – 270 – 18
= 7.812
3. Bentuk (x + p)(x – p) = x2 – p2
Untuk memahami bentuk ini, amati contoh berikut.
Contoh:
1. 21 × 19 = (20 + 1)(20 – 1)
= 202 – 12
= 400 – 1
= 399
2. 35 × 25 = (30 + 5)(30 – 5)
= 302 – 52
= 900 – 25
= 875
Contoh Soal & Pembahasan
Hitunglah dgn menggunakan bentuk-bentuk perkalian istimewa yg telah dijelaskan sebelumnya.
1. 5 × 45
Jawab:
5 × 45 = 5 × (40 + 5)
= (5 × 40) + (5 × 5)
= 200 + 25
= 225
2. 9 × 63
Jawab:
9 × 63 = 9 × (60 + 3)
= (9 × 60) + (9 × 3)
= 540 + 27
= 567
3. 8 × 72
Jawab:
8 × 72 = 8 × (70 + 2)
= (8 × 70) + (8 × 2)
= 560 + 16
= 576
4. 9 × 99
Jawab:
9 × 99 = 9 × (100 – 1)
= (9 × 100) – (9 × 1)
= 900 – 9
= 891
5. 64 × 63
Jawab:
64 × 63 = (60 + 4)(60 + 3)
= 602 + (4 + 3)60 + (4)(3)
= 3.600 + (7)60 + 12
= 3.600 + 420 + 12
= 4.032
6. 57 × 51
Jawab:
57 × 51 = (50 + 7)(50 + 1)
= 502 + (7 + 1)50 + (7)(1)
= 2.500 + (8)50 + 7
= 2.500 + 400 + 7
= 2.907
7. 83 × 89
Jawab:
83 × 89 = (80 + 3)(80 + 9)
= 802 + (3 + 9)80 + (3)(9)
= 6.400 + (11)80 + 27
= 6.400 + 880 + 27
= 7.307
8. 125 × 128
Jawab:
125 × 128 = (100 + 25)(100 + 28)
= 1002 + (25 + 28)100 + (25)(28)
= 10.000 + (53)100 + 700
= 10.000 + 5.300 + 700
= 16.000
Atau
125 × 128 = (120 + 5)(120 + 8)
= 1202 + (5 + 8)120 + (5)(8)
= 14.400 + 1.560 + 40
= 16.000
9. 31 × 29
Jawab:
31 × 29 = (30 + 1)(30 – 1)
= 302 – 12
= 900 – 1
= 899
10. 38 × 42
Jawab:
38 × 42 = (40 – 2)(40 + 2)
= 402 – 22
= 1.600 – 4
= 1.596
11. 52 × 38
Jawab:
52 × 38 = (50 + 2)(50 – 12)
= 502 + (2 – 12)50 + (2)(-12)
= 2.500 + (-10)50 – 24
= 2.500 – 500 – 24
= 1.976
12. 22 × 18
Jawab:
22 × 18 = (20 + 2)(20 – 2)
= 202 – 22
= 400 – 4
= 396
13. 32 × 28
Jawab:
32 × 28 = (30 + 2)(30 – 2)
= 302 – 22
= 900 – 4
= 896
14. 37 × 43
Jawab:
37 × 43 = (40 – 3)(40 + 3)
= 402 – 32
= 1.600 – 9
= 1.591
15. 61 × 59
Jawab:
61 × 59 = (60 + 1)(60 – 1)
= 602 – 12
= 3.600 – 1
= 3.599
16. 95 × 85
Jawab:
95 × 85 = (90 + 5)(90 – 5)
= 902 – 52
= 8.100 – 25
= 8.075
17. 23 × 17
Jawab:
23 × 17 = (20 + 3)(20 – 3)
= 202 – 32
= 400 – 9
= 391
18. 34 × 26
Jawab:
34 × 26 = (30 + 4)(30 – 4)
= 302 – 42
= 900 – 16
= 884
19. 39 × 41
Jawab:
39 × 41 = (40 – 1)(40 + 1)
= 402 – 12
= 1.600 – 1
= 1.599
20. 58 × 62
Jawab:
58 × 62 = (60 – 2)(60 + 2)
= 602 – 22
= 3.600 – 4
= 2.596