To determine which expressions are equivalent to [tex]\(\frac{c}{d}\)[/tex], we need to examine each option and see if it simplifies to the same value.
### Option A: [tex]\(\frac{-c}{-d}\)[/tex]
Here, both the numerator and the denominator have negative signs:
[tex]\[
\frac{-c}{-d}
\][/tex]
We can simplify this by factoring out the negative signs:
[tex]\[
\frac{-1 \cdot c}{-1 \cdot d} = \frac{(-1) \cdot c}{(-1) \cdot d} = \frac{c}{d}
\][/tex]
Hence, [tex]\(\frac{-c}{-d}\)[/tex] simplifies to [tex]\(\frac{c}{d}\)[/tex]. Therefore, Option A is equivalent to [tex]\(\frac{c}{d}\)[/tex].
### Option B: [tex]\(-\frac{c}{-d}\)[/tex]
Here, we have a negative sign in front of the fraction and a negative denominator:
[tex]\[
-\frac{c}{-d}
\][/tex]
The negative denominator can be brought into the numerator by multiplying the numerator by -1:
[tex]\[
-\frac{c}{-d} = - \left( \frac{c}{-d} \right) = - \left( \frac{c \cdot -1}{d \cdot -1} \right) = - \left( \frac{-c}{d} \right) = \frac{c}{d}
\][/tex]
Hence, [tex]\(-\frac{c}{-d}\)[/tex] simplifies to [tex]\(\frac{c}{d}\)[/tex]. Therefore, Option B is equivalent to [tex]\(\frac{c}{d}\)[/tex].
Given the above simplifications, both expressions are equivalent to [tex]\(\frac{c}{d}\)[/tex]. Therefore, the correct answers are:
- Option A: [tex]\(\frac{-c}{-d}\)[/tex]
- Option B: [tex]\(-\frac{c}{-d}\)[/tex]
