Answer :
Certainly! Let's find the square roots of these numbers by prime factorization method step-by-step:
### (i) 784
1. Factorize 784 into its prime factors:
- 784 ÷ 2 = 392
- 392 ÷ 2 = 196
- 196 ÷ 2 = 98
- 98 ÷ 2 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1
So, 784 = 2 × 2 × 2 × 2 × 7 × 7 = [tex]\( 2^4 \)[/tex] × [tex]\( 7^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 7^2 \)[/tex] )
3. Take one factor from each pair: 2 × 2 × 7 = 28
So, the square root of 784 is 28.
### (ii) 441
1. Factorize 441 into its prime factors:
- 441 ÷ 3 = 147
- 147 ÷ 3 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1
So, 441 = 3 × 3 × 7 × 7 = [tex]\( 3^2 \)[/tex] × [tex]\( 7^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 3^2 \)[/tex] ) × ( [tex]\( 7^2 \)[/tex] )
3. Take one factor from each pair: 3 × 7 = 21
So, the square root of 441 is 21.
### (iii) 1849
1. Factorize 1849 into its prime factors:
- 1849 ÷ 43 = 43
- 43 ÷ 43 = 1
So, 1849 = 43 × 43 = [tex]\( 43^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 43^2 \)[/tex] )
3. Take one factor from each pair: 43
So, the square root of 1849 is 43.
### (iv) 4356
1. Factorize 4356 into its prime factors:
- 4356 ÷ 2 = 2178
- 2178 ÷ 2 = 1089
- 1089 ÷ 3 = 363
- 363 ÷ 3 = 121
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1
So, 4356 = 2 × 2 × 3 × 3 × 11 × 11 = [tex]\( 2^2 \)[/tex] × [tex]\( 3^2 \)[/tex] × [tex]\( 11^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 3^2 \)[/tex] ) × ( [tex]\( 11^2 \)[/tex] )
3. Take one factor from each pair: 2 × 3 × 11 = 66
So, the square root of 4356 is 66.
### (vi) 8836
1. Factorize 8836 into its prime factors:
- 8836 ÷ 2 = 4418
- 4418 ÷ 2 = 2209
- 2209 ÷ 47 = 47
- 47 ÷ 47 = 1
So, 8836 = 2 × 2 × 47 × 47 = [tex]\( 2^2 \)[/tex] × [tex]\( 47^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 47^2 \)[/tex] )
3. Take one factor from each pair: 2 × 47 = 94
So, the square root of 8836 is 94.
### (vii) 8281
1. Factorize 8281 into its prime factors:
- 8281 ÷ 91 = 91
- 91 ÷ 7 = 13
- 13 ÷ 13 = 1
So, 8281 = 91 × 91 = [tex]\( 91^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 91^2 \)[/tex] )
3. Take one factor from each pair: 91
So, the square root of 8281 is 91.
### (viii) 9025
1. Factorize 9025 into its prime factors:
- 9025 ÷ 95 = 95
- 95 ÷ 5 = 19
- 19 ÷ 19 = 1
So, 9025 = 95 × 95 = [tex]\( 95^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 95^2 \)[/tex] )
3. Take one factor from each pair: 95
So, the square root of 9025 is 95.
### (v) 6241
1. Factorize 6241 into its prime factors:
- 6241 ÷ 79 = 79
- 79 ÷ 79 = 1
So, 6241 = 79 × 79 = [tex]\( 79^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 79^2 \)[/tex] )
3. Take one factor from each pair: 79
So, the square root of 6241 is 79.
In summary, here are the square roots of the given numbers:
1. [tex]\( \sqrt{784} = 28 \)[/tex]
2. [tex]\( \sqrt{441} = 21 \)[/tex]
3. [tex]\( \sqrt{1849} = 43 \)[/tex]
4. [tex]\( \sqrt{4356} = 66 \)[/tex]
5. [tex]\( \sqrt{8836} = 94 \)[/tex]
6. [tex]\( \sqrt{8281} = 91 \)[/tex]
7. [tex]\( \sqrt{9025} = 95 \)[/tex]
8. [tex]\( \sqrt{6241} = 79 \)[/tex]
### (i) 784
1. Factorize 784 into its prime factors:
- 784 ÷ 2 = 392
- 392 ÷ 2 = 196
- 196 ÷ 2 = 98
- 98 ÷ 2 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1
So, 784 = 2 × 2 × 2 × 2 × 7 × 7 = [tex]\( 2^4 \)[/tex] × [tex]\( 7^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 7^2 \)[/tex] )
3. Take one factor from each pair: 2 × 2 × 7 = 28
So, the square root of 784 is 28.
### (ii) 441
1. Factorize 441 into its prime factors:
- 441 ÷ 3 = 147
- 147 ÷ 3 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1
So, 441 = 3 × 3 × 7 × 7 = [tex]\( 3^2 \)[/tex] × [tex]\( 7^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 3^2 \)[/tex] ) × ( [tex]\( 7^2 \)[/tex] )
3. Take one factor from each pair: 3 × 7 = 21
So, the square root of 441 is 21.
### (iii) 1849
1. Factorize 1849 into its prime factors:
- 1849 ÷ 43 = 43
- 43 ÷ 43 = 1
So, 1849 = 43 × 43 = [tex]\( 43^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 43^2 \)[/tex] )
3. Take one factor from each pair: 43
So, the square root of 1849 is 43.
### (iv) 4356
1. Factorize 4356 into its prime factors:
- 4356 ÷ 2 = 2178
- 2178 ÷ 2 = 1089
- 1089 ÷ 3 = 363
- 363 ÷ 3 = 121
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1
So, 4356 = 2 × 2 × 3 × 3 × 11 × 11 = [tex]\( 2^2 \)[/tex] × [tex]\( 3^2 \)[/tex] × [tex]\( 11^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 3^2 \)[/tex] ) × ( [tex]\( 11^2 \)[/tex] )
3. Take one factor from each pair: 2 × 3 × 11 = 66
So, the square root of 4356 is 66.
### (vi) 8836
1. Factorize 8836 into its prime factors:
- 8836 ÷ 2 = 4418
- 4418 ÷ 2 = 2209
- 2209 ÷ 47 = 47
- 47 ÷ 47 = 1
So, 8836 = 2 × 2 × 47 × 47 = [tex]\( 2^2 \)[/tex] × [tex]\( 47^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 2^2 \)[/tex] ) × ( [tex]\( 47^2 \)[/tex] )
3. Take one factor from each pair: 2 × 47 = 94
So, the square root of 8836 is 94.
### (vii) 8281
1. Factorize 8281 into its prime factors:
- 8281 ÷ 91 = 91
- 91 ÷ 7 = 13
- 13 ÷ 13 = 1
So, 8281 = 91 × 91 = [tex]\( 91^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 91^2 \)[/tex] )
3. Take one factor from each pair: 91
So, the square root of 8281 is 91.
### (viii) 9025
1. Factorize 9025 into its prime factors:
- 9025 ÷ 95 = 95
- 95 ÷ 5 = 19
- 19 ÷ 19 = 1
So, 9025 = 95 × 95 = [tex]\( 95^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 95^2 \)[/tex] )
3. Take one factor from each pair: 95
So, the square root of 9025 is 95.
### (v) 6241
1. Factorize 6241 into its prime factors:
- 6241 ÷ 79 = 79
- 79 ÷ 79 = 1
So, 6241 = 79 × 79 = [tex]\( 79^2 \)[/tex]
2. Group the prime factors into pairs: ( [tex]\( 79^2 \)[/tex] )
3. Take one factor from each pair: 79
So, the square root of 6241 is 79.
In summary, here are the square roots of the given numbers:
1. [tex]\( \sqrt{784} = 28 \)[/tex]
2. [tex]\( \sqrt{441} = 21 \)[/tex]
3. [tex]\( \sqrt{1849} = 43 \)[/tex]
4. [tex]\( \sqrt{4356} = 66 \)[/tex]
5. [tex]\( \sqrt{8836} = 94 \)[/tex]
6. [tex]\( \sqrt{8281} = 91 \)[/tex]
7. [tex]\( \sqrt{9025} = 95 \)[/tex]
8. [tex]\( \sqrt{6241} = 79 \)[/tex]
