To determine which choices are equivalent to the quotient [tex]\( \frac{\sqrt{50}}{\sqrt{10}} \)[/tex], we will simplify each expression and compare the results.
Firstly, let's simplify the given quotient:
[tex]\[
\frac{\sqrt{50}}{\sqrt{10}}
\][/tex]
Using the property of square roots [tex]\(\sqrt{a}/\sqrt{b} = \sqrt{a/b}\)[/tex], we get:
[tex]\[
\frac{\sqrt{50}}{\sqrt{10}} = \sqrt{\frac{50}{10}} = \sqrt{5}
\][/tex]
So, the simplified form of the given quotient is [tex]\(\sqrt{5}\)[/tex].
Let's analyze each choice to see which ones are equivalent to [tex]\(\sqrt{5}\)[/tex]:
Choice A: [tex]\(\frac{15}{3}\)[/tex]
[tex]\[
\frac{15}{3} = 5
\][/tex]
This is clearly not equal to [tex]\(\sqrt{5}\)[/tex].
Choice B: 5
[tex]\[
5
\][/tex]
This is not equal to [tex]\(\sqrt{5}\)[/tex].
Choice C: [tex]\(\frac{\sqrt{25}}{\sqrt{5}}\)[/tex]
[tex]\[
\frac{\sqrt{25}}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}
\][/tex]
This is equal to [tex]\(\sqrt{5}\)[/tex].
Choice D: [tex]\(\sqrt{5}\)[/tex]
This is exactly [tex]\(\sqrt{5}\)[/tex].
Choice E: [tex]\(\sqrt{3}\)[/tex]
[tex]\[
\sqrt{3}
\][/tex]
This is not equal to [tex]\(\sqrt{5}\)[/tex].
Choice F: [tex]\(\frac{\sqrt{15}}{\sqrt{3}}\)[/tex]
[tex]\[
\frac{\sqrt{15}}{\sqrt{3}} = \sqrt{\frac{15}{3}} = \sqrt{5}
\][/tex]
This is equal to [tex]\(\sqrt{5}\)[/tex].
After evaluating all the choices, we find that the expressions equivalent to [tex]\(\frac{\sqrt{50}}{\sqrt{10}} = \sqrt{5}\)[/tex] are:
- Choice C: [tex]\(\frac{\sqrt{25}}{\sqrt{5}}\)[/tex]
- Choice D: [tex]\(\sqrt{5}\)[/tex]
- Choice F: [tex]\(\frac{\sqrt{15}}{\sqrt{3}}\)[/tex]
Thus, the choices that are equivalent to the quotient are:
C, D, and F.
