Select and use the identity that will help you to find the following products without multiplying directly:
- (i) (41)2
- (ii) (27)2
- (iii) (23 × 17)
- (iv) (135)2
- (v) (97)2
- (vi) (18 × 29)
- (vii) (34 × 43)
- (viii) (205)2
Step-by-step solution
Idea: Pick the identity that matches the shape of the product. A square of a number just above a round number → (a + b)2; just below → (a − b)2. Two numbers the same distance either side of a round number (23 and 17 are 20 ± 3) → (a + b)(a − b) = a2 − b2. Two numbers at different distances from the same round number → (x + a)(x + b).
(i) (41)2
- Identity: (a + b)2, since 41 = 40 + 1.
- (40 + 1)2 = 402 + 2(40)(1) + 12 = 1600 + 80 + 1 = 1681½ mark
(ii) (27)2
- Identity: (a − b)2, since 27 = 30 − 3 (30 is closer than 20).
- (30 − 3)2 = 302 − 2(30)(3) + 32 = 900 − 180 + 9 = 729½ mark
(iii) (23 × 17)
- Identity: (a + b)(a − b) = a2 − b2, since 23 = 20 + 3 and 17 = 20 − 3.
- 23 × 17 = (20 + 3)(20 − 3) = 202 − 32 = 400 − 9 = 391½ mark
(iv) (135)2
- Identity: (a + b)2, since 135 = 130 + 5 (1302 = 132 × 100 = 16900).
- (130 + 5)2 = 1302 + 2(130)(5) + 52 = 16900 + 1300 + 25 = 18225½ mark
(v) (97)2
- Identity: (a − b)2, since 97 = 100 − 3.
- (100 − 3)2 = 1002 − 2(100)(3) + 32 = 10000 − 600 + 9 = 9409½ mark
(vi) (18 × 29)
- Identity: (x + a)(x + b) = x2 + (a + b)x + ab. Both numbers are near 20: 18 = 20 + (−2) and 29 = 20 + 9. So x = 20, a = −2, b = 9.
- 18 × 29 = 202 + (−2 + 9)(20) + (−2)(9) = 400 + 140 − 18 = 522½ mark
- (Another route with the same identity: x = 30, a = −12, b = −1 gives 900 − 390 + 12 = 522.)
(vii) (34 × 43)
- Identity: (x + a)(x + b), with x = 40: 34 = 40 + (−6) and 43 = 40 + 3. So a = −6, b = 3.
- 34 × 43 = 402 + (−6 + 3)(40) + (−6)(3) = 1600 − 120 − 18 = 1462½ mark
(viii) (205)2
- Identity: (a + b)2, since 205 = 200 + 5.
- (200 + 5)2 = 2002 + 2(200)(5) + 52 = 40000 + 2000 + 25 = 42025½ mark
Check: Last digits: 3 × 7 = 21, so 23 × 17 ends in 1 ✓ (391). 8 × 9 = 72, so 18 × 29 ends in 2 ✓ (522). 4 × 3 = 12, so 34 × 43 ends in 2 ✓ (1462). Rough size: 34 × 43 is about 35 × 40 = 1400 ✓.
Answer to write in the exam
(i)
(41)2 = (40 + 1)2 = 402 + 2(40)(1) + 12 [(a + b)2 = a2 + 2ab + b2]
= 1600 + 80 + 1
∴ (41)2 = 1681
(ii)
(27)2 = (30 − 3)2 = 302 − 2(30)(3) + 32 [(a − b)2 = a2 − 2ab + b2]
= 900 − 180 + 9
∴ (27)2 = 729
(iii)
23 × 17 = (20 + 3)(20 − 3) = 202 − 32 [(a + b)(a − b) = a2 − b2]
= 400 − 9
∴ 23 × 17 = 391
(iv)
(135)2 = (130 + 5)2 = 1302 + 2(130)(5) + 52 [(a + b)2 = a2 + 2ab + b2]
= 16900 + 1300 + 25
∴ (135)2 = 18225
(v)
(97)2 = (100 − 3)2 = 1002 − 2(100)(3) + 32 [(a − b)2 = a2 − 2ab + b2]
= 10000 − 600 + 9
∴ (97)2 = 9409
(vi)
18 × 29 = [20 + (−2)](20 + 9) = 202 + (−2 + 9)(20) + (−2)(9) [(x + a)(x + b) = x2 + (a + b)x + ab]
= 400 + 140 − 18
∴ 18 × 29 = 522
(vii)
34 × 43 = [40 + (−6)](40 + 3) = 402 + (−6 + 3)(40) + (−6)(3) [(x + a)(x + b) = x2 + (a + b)x + ab]
= 1600 − 120 − 18
∴ 34 × 43 = 1462
(viii)
(205)2 = (200 + 5)2 = 2002 + 2(200)(5) + 52 [(a + b)2 = a2 + 2ab + b2]
= 40000 + 2000 + 25
∴ (205)2 = 42025
Common mistakes that cost marks
- In (iii), writing (20 + 3)(20 − 3) = 400 + 9. The identity gives a2 minus b2: 400 − 9 = 391.
- In (vi) and (vii), getting the sign of a or b wrong. 18 is 20 + (−2), so ab = (−2)(9) = −18, not +18.
- Using (a + b)(a − b) for 34 × 43. The numbers are not equally far from one round number (40 − 6 and 40 + 3), so use (x + a)(x + b) instead.
How this can come in the exam
The value of 102 × 98, found using a suitable identity, is
- 9996
- 10004
- 9994
- 9896
Show answer
(A) 9996
(100 + 2)(100 − 2) = 1002 − 22 = 10000 − 4 = 9996.
Using a suitable identity, find 47 × 53. Name the identity you used.
Show answer
47 = 50 − 3 and 53 = 50 + 3, so use (a + b)(a − b) = a2 − b2 (1 mark). 47 × 53 = 502 − 32 = 2500 − 9 = 2491 (1 mark).Try one yourself
Using a suitable identity, find 104 × 108 and (59)2.
Show answer
104 × 108 = (100 + 4)(100 + 8) = 10000 + 12 × 100 + 32 = 11232. (59)2 = (60 − 1)2 = 3600 − 120 + 1 = 3481.
More questions like this
- Factor the following:
- James and Reshma were talking about algebraic identities they learnt in school.
James: (a − b)2 (a + b) = (a2 − 2ab + b2)(a + b)
Reshma: I have a different idea. (a − b)2 (a + b) = (a − b) [(a − b) (a + b)] = (a − b)(a2 − b2)
I will find this product to get the answer.
According to you, who is correct and why?
Try to combine more such identities and find new results. - What do you think (a + b)3 will look like?
- What if we have a cube of edge a + b? Can we divide a cube of edge (a + b) into smaller cubes and cuboids and represent this new identity?
- What happens when we replace b with −b in this new identity?