# Square Roots

## Exercise 6.2

Question 1: Find the square of the following numbers.

(i) 32

Answer: 322 = 32 x 32 = 1024
But above method can be tough to calculate. It is easier to calculate such values in the following way:
Since, 32 can be written as (30+2)
So, 322 = (30+2)2 = (30+2)(30+2)
= 30(30+2)+2(30+2) = 302 + 30 x 2 + 2 x 30 + 22
= 900 + 60 + 60 + 4 = 1024

(ii) 35

Answer: (35)2 = (30+5)2 = (30+5)(30+5)
= 30(30+5)+5(30+5) = 302 + 30 x 5 + 5 x 30 + 52
= 900 + 150 + 150 + 25 = 1225

(iii) 86

Answer: 862 = (80 + 6)2 = (80 + 6)(80 + 6)
= 802 + 80 x 6 + 6 x 80 + 62
= 6400 + 480 + 480 + 36 = 7396

(iv) 93

Answer: 932 = (90+3)2 = (90 + 3) (90 + 3)
= 90 (90 + 3) + 3 (90 + 3) = 90 2 + 90 x 3 + 3 x 90 + 3 2
= 8100 + 270 + 270 + 9 = 8649

(v) 71

Answer: 71 2 = (70 + 1) 2 = (70 + 1) (70 + 1)
= 70 (70 + 1) + 1 (70 + 1) = 702 + 70 x 1 + 1 x 70 + 1 x 1
= 4900 + 70 + 70 + 1 = 4900 + 140 + 1 = 5040 + 1 = 5041

(vi) 46

Answer: 462 = (40+6)2 = (40 + 6) (40 + 6)
= 40 (40 + 6) + 6 (40 + 6) = 40 2 + 40 x 6 + 6 x 40 + 62
= 1600 + 240 + 240 + 36 = 1600 + 480 + 36 = 2080 + 36 = 2116

Question 2: Write a Pythagorean triplet whose one member is:

(i) 6

Answer: As we know 2m, m 2 + 1 and m2 - 1 form a Pythagorean triplet for any number, m > 1.
Let us assume 2m = 6
Therefore, m = 3
And, m2 + 1 = 3 2 + 1= 9 + 1 = 10
And, m 2 - 1 = 3 2 - 1 = 9 - 1 = 8
Test: 6 2 + 8 2 = 36 + 64 = 100 = 102
Hence, the triplet is 6, 8, and 10 Answer

(ii) 14

Answer: Let us assume, 2 m = 14, therefore, m = 7
Now, m 2 + 1 = 7 2 + 1 = 49 + 1 = 50
And, m 2 - 1 = 7 2 - 1 = 49 - 1 = 48
Test: 14 2 + 48 2 = 196 + 1304 = 2500 = 50 2
Hence, the triplet is 14, 48, and 50 Answer

(iii) 16

Answer: Let us assume 2 m = 16, then m = 8
Now, m 2 + 1 = 8 2 + 1 = 64 + 1 = 65
And, m 2 - 1 = 8 2 - 1 = 64 - 1 = 63
Test: 162 + 63 2 = 256 + 3969 = 4225 = 65 2
Hence, the triplet is 16, 63, and 65 Answer

(iv) 18

Answer: Let us assume 2 m = 18, therefore, m = 9
Now, m 2 + 1 = 9 2 + 1 = 81 + 1 = 82
And, m 2 - 1 = 9 2 - 1 = 81 - 1 = 80
Test: 18 2 + 80 2 = 324 + 6400 = 6724 = 82 2