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Which choices are equivalent to the expression below? Check all that apply.

[tex]\[ 6 \sqrt{3} \][/tex]

A. [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]

B. [tex]\(\sqrt{108}\)[/tex]

C. 108

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

E. [tex]\(\sqrt{54}\)[/tex]

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


Sagot :

Given the expression [tex]\(6 \sqrt{3}\)[/tex], let's verify which of the given choices are equivalent to it.

Choice A: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]

First, we simplify [tex]\(\sqrt{36}\)[/tex]:

[tex]\[ \sqrt{36} = 6 \][/tex]

So,

[tex]\[ \sqrt{3} \cdot \sqrt{36} = \sqrt{3} \cdot 6 = 6 \sqrt{3} \][/tex]

This matches the given expression. Therefore, Choice A is equivalent.

Choice B: [tex]\(\sqrt{108}\)[/tex]

Let's simplify [tex]\(\sqrt{108}\)[/tex]:

[tex]\[ 108 = 36 \cdot 3 \][/tex]

So,

[tex]\[ \sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \cdot \sqrt{3} = 6 \sqrt{3} \][/tex]

This also matches the given expression. Therefore, Choice B is equivalent.

Choice C: 108

The given expression is [tex]\(6 \sqrt{3}\)[/tex], which is clearly not equal to 108, as 108 is a scalar whereas [tex]\(6 \sqrt{3}\)[/tex] is an irrational number. Therefore, Choice C is not equivalent.

Choice D: [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]

Let's simplify [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]:

[tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2} \][/tex]

[tex]\[ \sqrt{6} = \sqrt{2 \cdot 3} = \sqrt{2} \cdot \sqrt{3} \][/tex]

So,

[tex]\[ \sqrt{18} \cdot \sqrt{6} = (3 \cdot \sqrt{2}) \cdot (\sqrt{2} \cdot \sqrt{3}) = 3 \cdot 2 \cdot \sqrt{3} = 6 \sqrt{3} \][/tex]

This matches the given expression. Therefore, Choice D is equivalent.

Choice E: [tex]\(\sqrt{54}\)[/tex]

Let's simplify [tex]\(\sqrt{54}\)[/tex]:

[tex]\[ 54 = 9 \cdot 6 \][/tex]

So,

[tex]\[ \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \cdot \sqrt{6} \][/tex]

This does not match the given expression [tex]\(6 \sqrt{3}\)[/tex]. Therefore, Choice E is not equivalent.

Choice F: [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]

Simplify [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]:

[tex]\[ \sqrt{3} \cdot \sqrt{6} = \sqrt{18} \][/tex]

And we have already simplified [tex]\(\sqrt{18}\)[/tex] in Choice D and found it to be [tex]\(3 \sqrt{2}\)[/tex], which does not match [tex]\(6 \sqrt{3}\)[/tex]. Therefore, Choice F is not equivalent.

In summary, the choices that are equivalent to [tex]\(6 \sqrt{3}\)[/tex] are:

- Choice A: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
- Choice B: [tex]\(\sqrt{108}\)[/tex]
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