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Sagot :
Sure! Let's determine the square roots via prime factorization for the given numbers:
### (i) 2916
1. Prime Factorization of 2916:
2916 ÷ 2 = 1458
1458 ÷ 2 = 729
729 ÷ 3 = 243
243 ÷ 3 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Therefore, the prime factors of 2916 are [tex]\(2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\)[/tex].
2. Group the factors in pairs:
[tex]\(2^2 \times 3^6\)[/tex]
3. Take the square root:
[tex]\(\sqrt{2916} = 2^{1} \times 3^{3} = 2 \times 27 = 54\)[/tex]
### (ii) 2704
1. Prime Factorization of 2704:
2704 ÷ 2 = 1352
1352 ÷ 2 = 676
676 ÷ 2 = 338
338 ÷ 2 = 169
169 ÷ 13 = 13
13 ÷ 13 = 1
Therefore, the prime factors of 2704 are [tex]\(2 \times 2 \times 2 \times 2 \times 13 \times 13\)[/tex].
2. Group the factors in pairs:
[tex]\(2^4 \times 13^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{2704} = 2^{2} \times 13^{1} = 4 \times 13 = 52\)[/tex]
### (iii) 7056
1. Prime Factorization of 7056:
7056 ÷ 2 = 3528
3528 ÷ 2 = 1764
1764 ÷ 2 = 882
882 ÷ 2 = 441
441 ÷ 3 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1
Therefore, the prime factors of 7056 are [tex]\(2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7\)[/tex].
2. Group the factors in pairs:
[tex]\(2^4 \times 3^2 \times 7^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{7056} = 2^{2} \times 3^{1} \times 7^{1} = 4 \times 3 \times 7 = 84\)[/tex]
### (iv) 9025
1. Prime Factorization of 9025:
9025 ÷ 5 = 1805
1805 ÷ 5 = 361
361 ÷ 19 = 19
19 ÷ 19 = 1
Therefore, the prime factors of 9025 are [tex]\(5 \times 5 \times 19 \times 19\)[/tex].
2. Group the factors in pairs:
[tex]\(5^2 \times 19^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{9025} = 5^{1} \times 19^{1} = 5 \times 19 = 95\)[/tex]
### (v) 9216
1. Prime Factorization of 9216:
9216 ÷ 2 = 4608
4608 ÷ 2 = 2304
2304 ÷ 2 = 1152
1152 ÷ 2 = 576
576 ÷ 2 = 288
288 ÷ 2 = 144
144 ÷ 2 = 72
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Therefore, the prime factors of 9216 are [tex]\(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\)[/tex].
2. Group the factors in pairs:
[tex]\(2^8 \times 3^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{9216} = 2^{4} \times 3^{1} = 16 \times 3 = 48 \times 2 = 96\)[/tex]
In conclusion, the square roots of the given numbers are:
- 2916: 54
- 2704: 52
- 7056: 84
- 9025: 95
- 9216: 96
### (i) 2916
1. Prime Factorization of 2916:
2916 ÷ 2 = 1458
1458 ÷ 2 = 729
729 ÷ 3 = 243
243 ÷ 3 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Therefore, the prime factors of 2916 are [tex]\(2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\)[/tex].
2. Group the factors in pairs:
[tex]\(2^2 \times 3^6\)[/tex]
3. Take the square root:
[tex]\(\sqrt{2916} = 2^{1} \times 3^{3} = 2 \times 27 = 54\)[/tex]
### (ii) 2704
1. Prime Factorization of 2704:
2704 ÷ 2 = 1352
1352 ÷ 2 = 676
676 ÷ 2 = 338
338 ÷ 2 = 169
169 ÷ 13 = 13
13 ÷ 13 = 1
Therefore, the prime factors of 2704 are [tex]\(2 \times 2 \times 2 \times 2 \times 13 \times 13\)[/tex].
2. Group the factors in pairs:
[tex]\(2^4 \times 13^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{2704} = 2^{2} \times 13^{1} = 4 \times 13 = 52\)[/tex]
### (iii) 7056
1. Prime Factorization of 7056:
7056 ÷ 2 = 3528
3528 ÷ 2 = 1764
1764 ÷ 2 = 882
882 ÷ 2 = 441
441 ÷ 3 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1
Therefore, the prime factors of 7056 are [tex]\(2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7\)[/tex].
2. Group the factors in pairs:
[tex]\(2^4 \times 3^2 \times 7^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{7056} = 2^{2} \times 3^{1} \times 7^{1} = 4 \times 3 \times 7 = 84\)[/tex]
### (iv) 9025
1. Prime Factorization of 9025:
9025 ÷ 5 = 1805
1805 ÷ 5 = 361
361 ÷ 19 = 19
19 ÷ 19 = 1
Therefore, the prime factors of 9025 are [tex]\(5 \times 5 \times 19 \times 19\)[/tex].
2. Group the factors in pairs:
[tex]\(5^2 \times 19^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{9025} = 5^{1} \times 19^{1} = 5 \times 19 = 95\)[/tex]
### (v) 9216
1. Prime Factorization of 9216:
9216 ÷ 2 = 4608
4608 ÷ 2 = 2304
2304 ÷ 2 = 1152
1152 ÷ 2 = 576
576 ÷ 2 = 288
288 ÷ 2 = 144
144 ÷ 2 = 72
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Therefore, the prime factors of 9216 are [tex]\(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\)[/tex].
2. Group the factors in pairs:
[tex]\(2^8 \times 3^2\)[/tex]
3. Take the square root:
[tex]\(\sqrt{9216} = 2^{4} \times 3^{1} = 16 \times 3 = 48 \times 2 = 96\)[/tex]
In conclusion, the square roots of the given numbers are:
- 2916: 54
- 2704: 52
- 7056: 84
- 9025: 95
- 9216: 96
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