To determine which choices are equivalent to the quotient
[tex]\[
\frac{\sqrt{100}}{\sqrt{20}},
\][/tex]
we will simplify and compare each option step by step.
First, let's simplify the given quotient:
[tex]\[
\frac{\sqrt{100}}{\sqrt{20}} = \frac{10}{\sqrt{20}}.
\][/tex]
Next, let's rationalize the denominator for clarity (optional).
[tex]\[
\frac{10}{\sqrt{20}} = \frac{10}{\sqrt{20}} \cdot \frac{\sqrt{20}}{\sqrt{20}} = \frac{10 \sqrt{20}}{20} = \frac{10 \sqrt{20}}{20} = \frac{\sqrt{20}}{2}.
\][/tex]
Since [tex]\(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}\)[/tex],
[tex]\[
\frac{2 \sqrt{5}}{2} = \sqrt{5}.
\][/tex]
So,
[tex]\[
\frac{\sqrt{100}}{\sqrt{20}} = \sqrt{5}.
\][/tex]
Now let's check the options one by one:
A. [tex]\(\frac{\sqrt{15}}{\sqrt{3}}\)[/tex]
Simplify:
[tex]\[
\frac{\sqrt{15}}{\sqrt{3}} = \sqrt{\frac{15}{3}} = \sqrt{5}.
\][/tex]
So, option A is equivalent to the quotient.
B. [tex]\(\frac{\sqrt{25}}{\sqrt{5}}\)[/tex]
Simplify:
[tex]\[
\frac{\sqrt{25}}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \frac{5}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{5 \sqrt{5}}{5} = \sqrt{5}.
\][/tex]
So, option B is equivalent to the quotient.
C. [tex]\(\sqrt{5}\)[/tex]
This is already simplified and it matches our quotient directly. So, option C is equivalent to the quotient.
D. 5
5 is a constant number and does not simplify to [tex]\(\sqrt{5}\)[/tex]. So, option D is not equivalent to the quotient.
E. [tex]\(\sqrt{3}\)[/tex]
[tex]\(\sqrt{3}\)[/tex] does not simplify to [tex]\(\sqrt{5}\)[/tex]. So, option E is not equivalent to the quotient.
F. [tex]\(\frac{15}{3}\)[/tex]
Simplify:
[tex]\[
\frac{15}{3} = 5.
\][/tex]
5 does not simplify to [tex]\(\sqrt{5}\)[/tex]. So, option F is not equivalent to the quotient.
Thus, the choices that are equivalent to the quotient [tex]\(\frac{\sqrt{100}}{\sqrt{20}}\)[/tex] are:
- A. [tex]\(\frac{\sqrt{15}}{\sqrt{3}}\)[/tex]
- B. [tex]\(\frac{\sqrt{25}}{\sqrt{5}}\)[/tex]
- C. [tex]\(\sqrt{5}\)[/tex]
The correct answers are:
[tex]\[
\boxed{A, B, C}
\][/tex]
