Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Which choices are equivalent to the expression below? Check all that apply. [tex]6 \sqrt{3}[/tex]

A. [tex]108[/tex]

B. [tex]\sqrt{54}[/tex]

C. [tex]\sqrt{3} \cdot \sqrt{36}[/tex]

D. [tex]\sqrt{18} \cdot \sqrt{6}[/tex]

E. [tex]\sqrt{108}[/tex]

F. [tex]\sqrt{3} \cdot \sqrt{6}[/tex]


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]