Answer:
[tex]0.357 = (357)/(1000)[/tex].
Step-by-step explanation:
A number is a rational number if and only if it is equal to the ratio between two integers. In other words, a number [tex]x[/tex] is a rational number if and only if there are integers (whole numbers) [tex]p[/tex] and [tex]q[/tex] ([tex]q \ne 0[/tex]) such that [tex]x = p / q[/tex].
Thus, the [tex]0.357[/tex] in this question would be a rational number as long as there are integers [tex]p[/tex] and [tex]q[/tex] ([tex]q \ne 0[/tex]) such that [tex]p / q = 0.357[/tex]. Simply finding the right [tex]p\![/tex] and [tex]q\1[/tex] would be sufficient for showing that [tex]0.357\![/tex] is rational.
Since [tex]0.357[/tex] is a terminating decimal, one possible way to find [tex]p\![/tex] and [tex]q\1[/tex] is to repeatedly multiply [tex]0.357\![/tex] by [tex]10[/tex] until a whole number is reached:
[tex]10 \times 0.357 = 3.57[/tex].
[tex]10^{2} \times 0.357 = 35.7[/tex].
[tex]10^{3} \times 0.357 = 357[/tex].
Thus, [tex]0.357 = (357) / (10^{3})[/tex], such that [tex]p = 357[/tex] and [tex]q = 1000[/tex] (both are integers, and [tex]q \ne 0[/tex]) would ensure that [tex]0.357 = p / q[/tex].
Therefore, [tex]0.357[/tex] is indeed a rational number.
