Answer:
The product of a repeating decimal (non-zero) and the square root of a non-perfect square would always be irrational.
Step-by-step explanation:
The sum and product between rational numbers are rational.
Fractions, finite decimals, and infinite repeating decimals are all rational numbers.
The square root of a perfect square is a rational number. However, the square root of a non-perfect square is not rational.
Let [tex]x[/tex] denote a repeating (non-zero) decimal ([tex]x \ne 0[/tex]). Let and [tex]y[/tex] denote the square root of a non-perfect square. Note that [tex]x\![/tex] would be a rational number, but [tex]y[/tex] would not be rational.
Assume the product [tex]x\, y[/tex] to be rational by contradiction. Since [tex]x \ne 0[/tex] and [tex]x[/tex] is a rational number, the multiplicative inverse [tex](1/x)[/tex] of [tex]x\![/tex] would be a rational number.
Left-multiply [tex]x\, y[/tex] by this multiplicative inverse to obtain: [tex](1/x) \, (x\, y) = ((1/x)\, x)\, y = y[/tex].
Since [tex](1/x)[/tex] is a rational number, and [tex]x\, y[/tex] is assumed to be rational, the product [tex](1/x) \, (x\, y) = y[/tex] should also be rational. This observation is a contradiction with the assumption that [tex]y[/tex] is not rational.
