Question

Which number produces an irrational number when multiplied by 1/3

a.0,777777...
b. square root of 27
c, 1/3
d. negative square root of 16

Answer 1

We can simply observe that.

1. 0.777... is rational, because it is a number with infinite but repeating decimal part.
2. 1/3 is rational, because it's the division between two integers
3. 
   `-√(16)=-4`, so this is rational as well.

Since the product of two rational numbers is always rational, we have that

`0.\bar{7}\cdot (1)/(3),\quad (1)/(3)\cdot (1)/(3),\quad -4\cdot (1)/(3)`

are all rationals, since they are the product of two rationals.

On the other hand, we have

`√(27)=√(9\cdot 3)=3√(3)`

and thus

`3√(3)\cdot (1)/(3)=√(3)`

which is irrational.