At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
