Question

find the reciprocal of \\(\frac{-1}{6}\\).

select the correct choice below and, if necessary, fill in the answer box within your choice.

a. the reciprocal of \\(\frac{-1}{6}\\) is \\(\square\\).

b. the reciprocal does not exist.

Explanation:

Step1: Recall the definition of reciprocal

The reciprocal of a number \( \frac{a}{b} \) (where \( a
eq0 \) and \( b
eq0 \)) is \( \frac{b}{a} \), because when we multiply a number by its reciprocal, the result is 1. Mathematically, \( \frac{a}{b}\times\frac{b}{a} = 1 \) (for \( a
eq0,b
eq0 \)).

Step2: Apply the definition to \( \frac{-1}{6} \)

For the number \( \frac{-1}{6} \), here \( a=-1 \) and \( b = 6 \). Using the reciprocal definition, the reciprocal should be \( \frac{6}{-1} \).

Step3: Simplify \( \frac{6}{-1} \)

Simplifying \( \frac{6}{-1} \) gives \( - 6 \), because dividing 6 by - 1 results in - 6. Also, we can check: \( \frac{-1}{6}\times(-6)=\frac{(-1)\times(-6)}{6}=\frac{6}{6} = 1 \), which satisfies the condition for reciprocal (product is 1).

Answer:

A. The reciprocal of \( \frac{-1}{6} \) is \( \boldsymbol{-6} \).