Question

assume that Assume a and b are nonzero rational numbers and c is an irrational number. For each of the following expressions, determine whether the result is irrational, rational, or both. Justify your answers.
a(b + c)
I think its irrational??

Answer 1

yep

Answer 2

Ok I am wondering how to explain that?

Answer 3

rational + irrational = irrational
and
constants * irrational you can think of as adding irrational to itself multiple times
for example 2*irrational = irrational+irrational
3*irrational = irrational + irrational + irrational

Answer 4

this fits into our first 3 rules we chose

Answer 5

irrational + irrational = irrational
rational + irrational = irrational
rational+rational = rational

Answer 6

just prove those 3 statements to yourself and then you can use those properties and time you want

Answer 7

I see.
thank you I was having a hard time understanding. Thank you so much

Answer 8

So if we throw a radical in there how does it change?

Answer 9

also another useful thing you will need for your next question
sqrt(prime) = irrational
the square root of a prime number is irrational
you can prove this by a contradiction
suppose
sqrt(prime) = rational
sqrt(prime)=a/b, where a and b is the lowest fraction possible meaning gcd(a,b)=1
then
prime = a^2/b^2
then
b^2prime=a^2
but this means there are an odd number of factors on the left and and even number of prime factors on the right this is a contradiction, ergo the srt of a prime is not rational but irrational

Answer 10

some of thoses steps are not obvious at first sight, you might beed to investigate some of those steps further to make sense of it

Answer 11

So this too would be irrational?