Which number produces an irrational number when multiplied by 0.4

Irrational number is a number without a perfect square root.

=2,3,5,6,7,8

Assuming 0.4 is in fact, 2 divided by 5 (and not an approximation of some number that would require an infinite amount of decimal places to exactly express), then multiplication by 0.4 is the same as multiplying by 2 and dividing by 5.

This means that multiplying any rational number by 2/5 will yield another rational number (as given x/y as an input, where x and y are integers, and multiplying it by 2/5 giving 2x/5y, will mean that 2x and 5y are also integers, and the result is also rational)

If we multiply an irrational number by 2/5, however, the result will be irrational.

So your answer is any number not in the set of rational numbers, multiplied by 0.4, will yield an irrational number.