**Question:**

Find the square root of each of the following by prime factorization.

(i) 441

(ii) 196

(iii) 529

(iv) 1764

(v) 1156

(vi) 4096

(vii) 7056

(viii) 8281

(ix) 11664

(x) 47089

(xi) 24336

(xii) 190969

(xiii) 586756

(xiv) 27225

(xv) 3013696

**Solution:**

(i) Resolving 441 into prime factors:

441 = 3 x 3 x 7 x 7

Grouping the factors into pairs of equal factors:

441 = (3 x 3) x (7 x 7)

Taking one factor for each pair, we get the square root of 441:

3 x 7 = 21

(ii) Resolving 196 into prime factors:

196 = 2 x 2 x 7 x 7

Grouping the factors into pairs of equal factors:

196 = (2 x 2) x (7 x 7)

Taking one factor for each pair, we get the square root of 196:

2 x 7 = 14

(iii) Resolving 529 into prime factors:

529 = 23 x 23

Grouping the factors into pairs of equal factors:

529= (23 x 23)

Taking one factor for each pair, we get the square root of 529 as 23.

(iv) Resolving 1764 into prime factors:

1764 = 2 x 2 x 3 x 3 x 7 x 7

Grouping the factors into pairs of equal factors:

1764 = (2 x 2) x (3 x 3) x (7 x 7)

Taking one factor for each pair, we get the square root of 1764:

2 x 3 x 7 = 42

(v) Resolving 1156 into prime factors:

1156 = 2 x 2 x 17 x 17

Grouping the factors into pairs of equal factors:

1156 = (2 x 2) x (17 x 17)

Taking one factor for each pair, we get the square root of 1156:

2 x 17 = 34

(vi) Resolving 4096 into prime factors:

4096 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Grouping the factors into pairs of equal factors:

4096 = (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2)

Taking one factor for each pair, we get the square root of 4096:

(2 x 2) x (2 x 2) x (2 x 2) = 64

(vii) Resolving 7056 into prime factors:

7056 = 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7

Grouping the factors into pairs of equal factors:

7056 = (2 x 2) x (2 x 2) x (3 x 3) x (7 x 7)

Taking one factor for each pair, we get the square root of 705:

2 x 2 x 3 x 7 = 84

(viii) Resolving 8281 into prime factors:

8281 = 7 x 7 x 13 x 13

Grouping the factors into pairs of equal factors:

8281 = (7 x 7) x (13 x 13)

Taking one factor for each pair, we get the square root of 8281:

7 x 13 = 91

(ix) Resolving 11664 into prime factors:

11664 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3

Grouping the factors into pairs of equal factors:

11664 = (2 x 2) x (2 x 2) x (3 x 3) x (3 x 3) x (3 x 3)

Taking one factor for each pair, we get the square root of 11664:

2 x 2 x 3 x 3 x 3 = 108

(x) Resolving 47089 into prime factors:

47089 = 7 x 7 x 31 x 31

Grouping the factors into pairs of equal factors:

47089 = (7 x 7) x (31 x 31)

Taking one factor for each pair, we get the square root of 47089:

7 x 31 = 217

(xi) Resolving 24336 into prime factors:

24336 = 2 x 2 x 2 x 2 x 3 x 3 x 13 x 13

Grouping the factors into pairs of equal factors:

24336 = (2 x 2) x (2 x 2) x (3 x 3) x (13 x 13)

Taking one factor for each pair, we get the square root of 24336:

2 x 2 x 3 x 13 = 156

(xii) Resolving 190969 into prime factors:

190969 = 19 x 19 x 23 x 23

Grouping the factors into pairs of equal factors:

190969 = (19 x 19) x (23 x 23)

Taking one factor for each pair, we get the square root of 190969:

19 x 23 = 437

(xiii) Resolving 586756 into prime factors:

586756 = 2 x 2 x 383 x 383

Grouping the factors into pairs of equal factors:

586756 = (2 x 2) x (383 x 383)

Taking one factor for each pair, we get the square root of 586756:

2 x 383 = 766

(xiv) Resolving 27225 into prime factors:

27225 = 3 x 3 x 5 x 5 x 11 x 11

Grouping the factors into pairs of equal factors:

27225 = (3 x 3) x (5 x 5) x (11 x 11)

Taking one factor for each pair, we get the square root of 27225:

3 x 5 x 11 = 165

(xv) Resolving 3013696 into prime factors:

3013696 = 2 x 2 x 2 x 2 x 2 x 2 x 7 x 7 x 31 x 31

Grouping the factors into pairs of equal factors:

3013696 = (2 x 2) x (2 x 2) x (2 x 2) x (7 x 7) x (31 x 31)

Taking one factor for each pair, we get the square root of 3013696:

2 x 2 x 2 x 7 x 31 = 1736