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Which of the following expressions are equivalent to [tex]\frac{c}{d}[/tex]?

Choose all answers that apply:

A. [tex]\frac{-c}{-d}[/tex]

B. [tex]-\frac{c}{-d}[/tex]

C. None of the above

Sagot :

GinnyAnsweridn
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]
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