Question

A student is attempting to show that the product of a nonzero rational number and an irrational number is always rational or always irrational.

Answer 1

Part A: Find the value of \[0.22 \] * \[\sqrt{112}\]and place it in simplified form.

Answer 2

Part B: Is the answer in Part A irrational or rational? Make a conjecture about the product of a nonzero rational number and an irrational number.

Answer 3

not really clear what the question is
find the value of what?

Answer 4

I need help with part B the answer to A is 2.32826115

Answer 5

it just says "make a conjecture" but the truth is that a rational number times an irrational number is always irrational

Answer 6

the proof is very simple if you want to see it

Answer 7

yes please

Answer 8

a rational number is any number that can be expressed as a fraction (ratio of two integers)
an irrational number is one that cannot be expressed that way
we need to know this to start

Answer 9

perhaps the easiest way to show that the product of an irrational number and a rational number is irrational is to prove it by contradiction, that is, assume that the product IS rational, then get a contradiction

Answer 10

so suppose \(a\) is rational, \(b\) is irrational and \(ab=c\) is rational
that means that \(b=\frac{c}{a}\) is irrational, but since both \(c\) and \(a\) are rational so is \(\frac{c}{a}\) contradicting the fact that \(b\) if irrational

Answer 11

btw if this seems like kicking the can down the road, it is not
you can prove for yourself that if \(c, a\) are rational, then so is \(\frac{c}{a}\) by writing them both as the ratio of two integers, invert and multiply to get another ratio of two integers

Answer 12

thank you