1. From your description, I can infer that the multiplication is:
The first thing we are going to do is simplify the radicands 196 ans 108 (picture 1):
and
Knowing this, we can rewrite our radicals as follows:
Remember that
; in other words if the radicand is raised to the same power as the index of the radical, we can take the radicand out. Since 2 and 7 are raised to the power 2 and the index of the radical is also 2 (square root), we can take out 2 and 7:
Look! we have the same numerator and denominator in our fraction, so we can cancel them both:
Notice that we can write
as
, so we can rewrite our expression one last time:
We can conclude that
the correct option is:
2. The <span>product of a nonzero rational number and an irrational number is always an
irrational number.
Proof by contradiction:
Lets assume that the product of an irrational number and a rational non-zero number is always rational.
Let </span>
be and irrational number and let
and
be two rational numbers with
,
,
, and
are non-zero integers.
Since integers are closed under multiplication,
is a rational number. Since
is an irrational number and
, we have a logical contradiction, so we can conclude that the product of an irrational number and a rational non-zero number is always an
irrational number.