Find the square of the following numbers.
(i) 32 (ii) 35 (iii) 86
(iv) 93 (v) 71 (vi) 46
(i) (32)2 = (30 + 2)2
= 302 + 2(30)(2)+(2)2
= 900 + 120 + 4
= 1024
(i) (35)2 = (30 + 5)2
= 302 + 2(30)(5)+(5)2
= 900 + 300 + 25
= 1225
Second method
(35)2 = 3 x (3 + 1) x 100 + 25
= 3 x 4 x 100 + 25
= 1200 + 25
= 1225\
(iii) (86)2 = (80 + 6)2
= (80)2 + 2(80)(6)+(6)2
= 6400 + 960 + 9
= 8649
(iv) (93)2 = (90 + 3)2
= (90)2 + 2(90)(3)+(3)2
= 8100 + 540 + 9
= 8649
(v) (71)2 = (70 + 1)2
= (70)2 + 2(70)(1)+(1)2
= 4900 + 140 + 1
= 5041
(vi) (46)2 = (40 + 6)2
= (40)2 + 2(40)(6)+(6)2
= 1600 + 480 + 36
= 2116
Write a Pytagorean triplet whose one member is.
(i) 6 (ii) 14 (iii) 16 (iv) 18
(i) Let 2n = 6 ∴ n = 3
Now, n2 -1 = 32 - 1 = 8
and n2 +1 = 32 + 1 = 10
Thus the required Pythagorean triplet is 6,8,10.
(ii) Let 2n = 14 ∴ n = 7
Now, n2 -1 = 72 - 1 = 48
and n2 +1 = 72 + 1 = 50
Thus the required Pythagorean triplet is 14,48,50.
(iii) Let 2n = 16 ∴ n = 8
Now, n2 -1 = 82 - 1 = 63
and n2 +1 = 82 + 1 = 65
∴ The required Pythagorean triplet is 16,63,65.
(iv) Let 2n = 18 ∴ n = 9
Now, n2 -1 = 92 - 1 = 80
and n2 +1 = 92 + 1 = 82
∴ The required Pythagorean triplet is 18,80,82.
Find the square of the following numbers containing 5 in unit’s place.
(i) 15 (ii) 95 (iii) 105 (iv) 205
(i) (15)2 = 1 x (1 + 1) x 100 + 25
= 1 x 2 x 100 + 25
= 200 + 25 = 225
(ii) (95)2 = 9 x (9 + 1) x 100 + 25
= 9 x 10 x 100 + 25
= 9000 + 25 = 9025
(iii) (105)2 = 10 x (10 + 1) x 100 + 25
= 10 x 11 x 100 + 25
= 11000 + 25 = 11025
(iv) (205)2 = 20 x (20 + 1) x 100 + 25
= 20 x 21 x 100 + 25
= 42000 + 25 = 42025
(–1) = 1. Is –1, a square root of 1?
(–2)2 = 4. Is –2, a square root of 4?
(–9)2 = 81. Is –9, a square root of 81?
(i) Since (–1) × (–1) = 1
i.e. (–1)2 = 1
∴ Square root of 1 can also be –1.
Similarly,
(ii) Yes (–2) is a square root of 4.
(iii) Yes (–9) is a square root of 81.
Since we have to consider the positive square roots only and the symbol for a positive square root is .
∴ = 4 (and not (-4)
Similarly, means, the positive square root of 25 i.e. 5.
We can also find the square root by subtracting successive odd numbers starting from 1.