Question

find the reciprocal of the number, if it exists.
\\(\frac{45}{-17}\\)

select the correct choice below and fill in the answer box within your choice.
\\(\bigcirc\\) a. the reciprocal does not exist.
\\(\bigcirc\\) b.
the reciprocal of \\(\frac{45}{-17}\\) is \\(\square\\). (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the definition of reciprocal

The reciprocal of a non - zero number \(a\) (where \(a=\frac{m}{n},m
eq0,n
eq0\)) is \(\frac{n}{m}\), because the product of a number and its reciprocal is \(1\), i.e., \(a\times\frac{n}{m}=\frac{m}{n}\times\frac{n}{m} = 1\) (for \(m
eq0,n
eq0\)).

Step2: Find the reciprocal of \(\frac{45}{- 17}\)

Given the number \(\frac{45}{-17}\), we can rewrite it as \(-\frac{45}{17}\). Since \(45
eq0\) and \(- 17
eq0\), the reciprocal of \(\frac{45}{-17}\) (or \(-\frac{45}{17}\)) is obtained by interchanging the numerator and the denominator. So the reciprocal is \(\frac{-17}{45}=-\frac{17}{45}\). Also, we can check: \(\frac{45}{-17}\times\frac{-17}{45}=\frac{45\times(- 17)}{-17\times45}=1\), which satisfies the definition of reciprocal.

Answer:

B. The reciprocal of \(\frac{45}{-17}\) is \(-\frac{17}{45}\)