Write the prime factorization of the following numbers and hence find their square roots.

Solution:

(i) The prime factorisation of 7744:

$7744=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11$

Grouping them into pairs of equal factors, we get:

$7744=(2 \times 2) \times(2 \times 2) \times(2 \times 2) \times(11 \times 11)$

Taking one factor from each pair, we get :

$\sqrt{7744}=2 \times 2 \times 2 \times 2 \times 11=88$

(ii) The prime factorisation of 9604:

$9604=2 \times 2 \times 7 \times 7 \times 7 \times 7$

Grouping them into pairs of equal factors, we get:

$9604=(2 \times 2) \times(7 \times 7) \times(7 \times 7)$

Taking one factor from each pair, we get:

$\sqrt{9604}=2 \times 7 \times 7=98$

(iii) The prime factorisation of 5929:

$5929=7 \times 7 \times 11 \times 11$

Grouping them into pairs of equal factors, we get:

$5929=(7 \times 7) \times(11 \times 11)$

Taking one factor from each pair, we get:

$\sqrt{5929}=7 \times 11=77$

(iv) The prime factorisation of 7056:

$7056=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7$

Grouping them into pairs of equal factors, we get:

$7056=(2 \times 2) \times(2 \times 2) \times(3 \times 3) \times(7 \times 7)$

Taking one factor from each pair, we get:

$\sqrt{7056}=2 \times 2 \times 3 \times 7=84$