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Sagot :
Absolutely! Let's go through each fraction and mixed number step-by-step to find their square roots:
### 1. Square root of [tex]\(\frac{4}{81}\)[/tex]
We need the square root of both the numerator and the denominator separately:
- [tex]\(\sqrt{4} = 2\)[/tex]
- [tex]\(\sqrt{81} = 9\)[/tex]
Thus, the square root of [tex]\(\frac{4}{81}\)[/tex] is [tex]\(\frac{2}{9}\)[/tex].
### 2. Square root of [tex]\(\frac{64}{289}\)[/tex]
Again, find the square root of both the numerator and the denominator:
- [tex]\(\sqrt{64} = 8\)[/tex]
- [tex]\(\sqrt{289} = 17\)[/tex]
So, the square root of [tex]\(\frac{64}{289}\)[/tex] is [tex]\(\frac{8}{17}\)[/tex].
### 3. Square root of 5
A whole number can be expressed as a fraction over 1 ([tex]\(\frac{5}{1}\)[/tex]):
- [tex]\(\sqrt{5} \approx 2.236\)[/tex]
- [tex]\(\sqrt{1} = 1\)[/tex]
Therefore, the square root of 5 is approximately [tex]\(\frac{2.236}{1}\)[/tex] or simply [tex]\(2.236\)[/tex].
### 4. Square root of [tex]\(3 \frac{97}{144}\)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 3 \frac{97}{144} = \frac{3 \times 144 + 97}{144} = \frac{529}{144} \][/tex]
Now, find the square roots:
- [tex]\(\sqrt{529} = 23\)[/tex]
- [tex]\(\sqrt{144} = 12\)[/tex]
So, the square root of [tex]\(3 \frac{97}{144}\)[/tex] is [tex]\(\frac{23}{12}\)[/tex].
### 5. Square root of [tex]\(2 \frac{7}{9}\)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{7}{9} = \frac{2 \times 9 + 7}{9} = \frac{25}{9} \][/tex]
Then, find the square roots:
- [tex]\(\sqrt{25} = 5\)[/tex]
- [tex]\(\sqrt{9} = 3\)[/tex]
Thus, the square root of [tex]\(2 \frac{7}{9}\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].
### 6. Square root of 5 (again)
As in step 3, the square root of 5 is approximately [tex]\(\frac{2.236}{1}\)[/tex] or [tex]\(2.236\)[/tex].
### 7. Square root of [tex]\(1 \frac{155}{169}\)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 1 \frac{155}{169} = \frac{1 \times 169 + 155}{169} = \frac{324}{169} \][/tex]
And find the square roots:
- [tex]\(\sqrt{324} = 18\)[/tex]
- [tex]\(\sqrt{169} = 13\)[/tex]
Therefore, the square root of [tex]\(1 \frac{155}{169}\)[/tex] is [tex]\(\frac{18}{13}\)[/tex].
### 8. Square root of [tex]\(7 \frac{106}{225}\)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 7 \frac{106}{225} = \frac{7 \times 225 + 106}{225} = \frac{1681}{225} \][/tex]
Find the square roots:
- [tex]\(\sqrt{1681} = 41\)[/tex]
- [tex]\(\sqrt{225} = 15\)[/tex]
Thus, the square root of [tex]\(7 \frac{106}{225}\)[/tex] is [tex]\(\frac{41}{15}\)[/tex].
### 9. Square root of 1
A whole number 1 can be expressed as [tex]\(\frac{1}{1}\)[/tex]:
- [tex]\(\sqrt{1} = 1\)[/tex]
- [tex]\(\sqrt{1} = 1\)[/tex]
Hence, the square root of 1 is [tex]\(\frac{1}{1}\)[/tex] or simply [tex]\(1\)[/tex].
### Summary of Results
The square roots of the given fractions and mixed numbers are:
1. [tex]\(\frac{2}{9}\)[/tex]
2. [tex]\(\frac{8}{17}\)[/tex]
3. [tex]\(2.236\)[/tex]
4. [tex]\(\frac{23}{12}\)[/tex]
5. [tex]\(\frac{5}{3}\)[/tex]
6. [tex]\(2.236\)[/tex]
7. [tex]\(\frac{18}{13}\)[/tex]
8. [tex]\(\frac{41}{15}\)[/tex]
9. [tex]\(1\)[/tex]
These calculations provide us with the square roots required for each specified fraction and mixed number.
### 1. Square root of [tex]\(\frac{4}{81}\)[/tex]
We need the square root of both the numerator and the denominator separately:
- [tex]\(\sqrt{4} = 2\)[/tex]
- [tex]\(\sqrt{81} = 9\)[/tex]
Thus, the square root of [tex]\(\frac{4}{81}\)[/tex] is [tex]\(\frac{2}{9}\)[/tex].
### 2. Square root of [tex]\(\frac{64}{289}\)[/tex]
Again, find the square root of both the numerator and the denominator:
- [tex]\(\sqrt{64} = 8\)[/tex]
- [tex]\(\sqrt{289} = 17\)[/tex]
So, the square root of [tex]\(\frac{64}{289}\)[/tex] is [tex]\(\frac{8}{17}\)[/tex].
### 3. Square root of 5
A whole number can be expressed as a fraction over 1 ([tex]\(\frac{5}{1}\)[/tex]):
- [tex]\(\sqrt{5} \approx 2.236\)[/tex]
- [tex]\(\sqrt{1} = 1\)[/tex]
Therefore, the square root of 5 is approximately [tex]\(\frac{2.236}{1}\)[/tex] or simply [tex]\(2.236\)[/tex].
### 4. Square root of [tex]\(3 \frac{97}{144}\)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 3 \frac{97}{144} = \frac{3 \times 144 + 97}{144} = \frac{529}{144} \][/tex]
Now, find the square roots:
- [tex]\(\sqrt{529} = 23\)[/tex]
- [tex]\(\sqrt{144} = 12\)[/tex]
So, the square root of [tex]\(3 \frac{97}{144}\)[/tex] is [tex]\(\frac{23}{12}\)[/tex].
### 5. Square root of [tex]\(2 \frac{7}{9}\)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{7}{9} = \frac{2 \times 9 + 7}{9} = \frac{25}{9} \][/tex]
Then, find the square roots:
- [tex]\(\sqrt{25} = 5\)[/tex]
- [tex]\(\sqrt{9} = 3\)[/tex]
Thus, the square root of [tex]\(2 \frac{7}{9}\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].
### 6. Square root of 5 (again)
As in step 3, the square root of 5 is approximately [tex]\(\frac{2.236}{1}\)[/tex] or [tex]\(2.236\)[/tex].
### 7. Square root of [tex]\(1 \frac{155}{169}\)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 1 \frac{155}{169} = \frac{1 \times 169 + 155}{169} = \frac{324}{169} \][/tex]
And find the square roots:
- [tex]\(\sqrt{324} = 18\)[/tex]
- [tex]\(\sqrt{169} = 13\)[/tex]
Therefore, the square root of [tex]\(1 \frac{155}{169}\)[/tex] is [tex]\(\frac{18}{13}\)[/tex].
### 8. Square root of [tex]\(7 \frac{106}{225}\)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 7 \frac{106}{225} = \frac{7 \times 225 + 106}{225} = \frac{1681}{225} \][/tex]
Find the square roots:
- [tex]\(\sqrt{1681} = 41\)[/tex]
- [tex]\(\sqrt{225} = 15\)[/tex]
Thus, the square root of [tex]\(7 \frac{106}{225}\)[/tex] is [tex]\(\frac{41}{15}\)[/tex].
### 9. Square root of 1
A whole number 1 can be expressed as [tex]\(\frac{1}{1}\)[/tex]:
- [tex]\(\sqrt{1} = 1\)[/tex]
- [tex]\(\sqrt{1} = 1\)[/tex]
Hence, the square root of 1 is [tex]\(\frac{1}{1}\)[/tex] or simply [tex]\(1\)[/tex].
### Summary of Results
The square roots of the given fractions and mixed numbers are:
1. [tex]\(\frac{2}{9}\)[/tex]
2. [tex]\(\frac{8}{17}\)[/tex]
3. [tex]\(2.236\)[/tex]
4. [tex]\(\frac{23}{12}\)[/tex]
5. [tex]\(\frac{5}{3}\)[/tex]
6. [tex]\(2.236\)[/tex]
7. [tex]\(\frac{18}{13}\)[/tex]
8. [tex]\(\frac{41}{15}\)[/tex]
9. [tex]\(1\)[/tex]
These calculations provide us with the square roots required for each specified fraction and mixed number.
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