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Sagot :
To determine which choices are equivalent to the expression [tex]\(6 \sqrt{3}\)[/tex], let's evaluate each of the given options step-by-step:
### Choice A: [tex]\(108\)[/tex]
- This is a simple integer value.
- To check if it equals [tex]\(6 \sqrt{3}\)[/tex]:
[tex]\[6 \sqrt{3} \approx 6 \cdot 1.732 \approx 10.392\][/tex]
- Clearly, [tex]\(108\)[/tex] is much larger than [tex]\(10.392\)[/tex], so this is not equivalent.
### Choice B: [tex]\(\sqrt{54}\)[/tex]
- Simplifying [tex]\(\sqrt{54}\)[/tex]:
[tex]\[\sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6}\][/tex]
- To see if this matches [tex]\(6 \sqrt{3}\)[/tex], let's note that:
[tex]\[3 \sqrt{6}\][/tex]
[tex]\[3 \cdot (\sqrt{3} \cdot \sqrt{2}) = 3 \cdot \sqrt{6}\][/tex]
- Comparing this to [tex]\(6 \sqrt{3}\)[/tex], these are not quite the same since the multiplication by [tex]\(\sqrt{2}\)[/tex] means it's larger than [tex]\(6 \sqrt{3}\)[/tex].
### Choice C: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
- Simplifying [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]:
[tex]\[\sqrt{3} \cdot \sqrt{36} = \sqrt{3} \cdot 6 = 6 \sqrt{3}\][/tex]
- This is exactly the same as [tex]\(6 \sqrt{3}\)[/tex].
### Choice D: [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]
- Simplifying [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]:
[tex]\[\sqrt{18 \cdot 6} = \sqrt{108}\][/tex]
[tex]\[\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}\][/tex]
- This expression simplifies to [tex]\(6 \sqrt{3}\)[/tex].
### Choice E: [tex]\(\sqrt{108}\)[/tex]
- Simplifying [tex]\(\sqrt{108}\)[/tex]:
[tex]\[\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}\][/tex]
- This is again exactly equal to [tex]\(6 \sqrt{3}\)[/tex].
### Choice F: [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]
- Simplifying [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]:
[tex]\[\sqrt{3} \cdot \sqrt{6} = \sqrt{18}\][/tex]
- To check equivalence:
[tex]\[\sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}\][/tex]
[tex]\[3 \sqrt{2}\][/tex]
[tex]\[3 \cdot 1.414 \approx 4.242\][/tex]
- This is smaller than [tex]\(6 \sqrt{3}\)[/tex], so it is not equivalent.
Summarizing the above analyses, the choices that are equivalent to [tex]\(6 \sqrt{3}\)[/tex] are:
- Choice C: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
- Choice D: [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]
- Choice E: [tex]\(\sqrt{108}\)[/tex]
### Choice A: [tex]\(108\)[/tex]
- This is a simple integer value.
- To check if it equals [tex]\(6 \sqrt{3}\)[/tex]:
[tex]\[6 \sqrt{3} \approx 6 \cdot 1.732 \approx 10.392\][/tex]
- Clearly, [tex]\(108\)[/tex] is much larger than [tex]\(10.392\)[/tex], so this is not equivalent.
### Choice B: [tex]\(\sqrt{54}\)[/tex]
- Simplifying [tex]\(\sqrt{54}\)[/tex]:
[tex]\[\sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6}\][/tex]
- To see if this matches [tex]\(6 \sqrt{3}\)[/tex], let's note that:
[tex]\[3 \sqrt{6}\][/tex]
[tex]\[3 \cdot (\sqrt{3} \cdot \sqrt{2}) = 3 \cdot \sqrt{6}\][/tex]
- Comparing this to [tex]\(6 \sqrt{3}\)[/tex], these are not quite the same since the multiplication by [tex]\(\sqrt{2}\)[/tex] means it's larger than [tex]\(6 \sqrt{3}\)[/tex].
### Choice C: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
- Simplifying [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]:
[tex]\[\sqrt{3} \cdot \sqrt{36} = \sqrt{3} \cdot 6 = 6 \sqrt{3}\][/tex]
- This is exactly the same as [tex]\(6 \sqrt{3}\)[/tex].
### Choice D: [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]
- Simplifying [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]:
[tex]\[\sqrt{18 \cdot 6} = \sqrt{108}\][/tex]
[tex]\[\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}\][/tex]
- This expression simplifies to [tex]\(6 \sqrt{3}\)[/tex].
### Choice E: [tex]\(\sqrt{108}\)[/tex]
- Simplifying [tex]\(\sqrt{108}\)[/tex]:
[tex]\[\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}\][/tex]
- This is again exactly equal to [tex]\(6 \sqrt{3}\)[/tex].
### Choice F: [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]
- Simplifying [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]:
[tex]\[\sqrt{3} \cdot \sqrt{6} = \sqrt{18}\][/tex]
- To check equivalence:
[tex]\[\sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}\][/tex]
[tex]\[3 \sqrt{2}\][/tex]
[tex]\[3 \cdot 1.414 \approx 4.242\][/tex]
- This is smaller than [tex]\(6 \sqrt{3}\)[/tex], so it is not equivalent.
Summarizing the above analyses, the choices that are equivalent to [tex]\(6 \sqrt{3}\)[/tex] are:
- Choice C: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
- Choice D: [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]
- Choice E: [tex]\(\sqrt{108}\)[/tex]
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