To determine which choices are equivalent to the quotient [tex]\( \frac{\sqrt{75}}{\sqrt{15}} \)[/tex], we can simplify the expression step-by-step.
First, we start with the given expression:
[tex]\[ \frac{\sqrt{75}}{\sqrt{15}} \][/tex]
We know from the properties of square roots that:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \][/tex]
Applying this property, we get:
[tex]\[ \frac{\sqrt{75}}{\sqrt{15}} = \sqrt{\frac{75}{15}} \][/tex]
Next, simplify the fraction inside the square root:
[tex]\[ \frac{75}{15} = 5 \][/tex]
Thus, the expression reduces to:
[tex]\[ \sqrt{5} \][/tex]
So, [tex]\(\frac{\sqrt{75}}{\sqrt{15}}\)[/tex] simplifies to [tex]\(\sqrt{5}\)[/tex].
Now, let's evaluate which of the given choices are equivalent to [tex]\(\sqrt{5}\)[/tex]:
A. [tex]\( \frac{\sqrt{25}}{\sqrt{5}} \)[/tex]
[tex]\[ \frac{\sqrt{25}}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5} \][/tex]
This is equivalent to [tex]\( \sqrt{5} \)[/tex].
B. [tex]\( 5 \)[/tex]
This is not equivalent to [tex]\( \sqrt{5} \)[/tex].
C. [tex]\( \sqrt{5} \)[/tex]
This is obviously equivalent to [tex]\( \sqrt{5} \)[/tex].
D. [tex]\( \frac{15}{3} \)[/tex]
[tex]\[ \frac{15}{3} = 5 \][/tex]
This is not equivalent to [tex]\( \sqrt{5} \)[/tex].
E. [tex]\( \frac{\sqrt{15}}{\sqrt{3}} \)[/tex]
[tex]\[ \frac{\sqrt{15}}{\sqrt{3}} = \sqrt{\frac{15}{3}} = \sqrt{5} \][/tex]
This is equivalent to [tex]\( \sqrt{5} \)[/tex].
F. [tex]\( \sqrt{3} \)[/tex]
This is not equivalent to [tex]\( \sqrt{5} \)[/tex].
Therefore, the choices that are equivalent to [tex]\( \frac{\sqrt{75}}{\sqrt{15}} \)[/tex] are:
[tex]\[ \text{A, C, and E} \][/tex]
These correspond to the numbers:
[tex]\[ 1, 3, and 5 \][/tex]
Thus, the choices are:
[tex]\[ \boxed{1, 3, 5} \][/tex]
