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

1. shaniehh

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

2. shaniehh

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.

3. anonymous

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

4. shaniehh

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

5. anonymous

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

6. anonymous

the proof is very simple if you want to see it

7. shaniehh

8. anonymous

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

9. anonymous

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

10. anonymous

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

11. anonymous

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

12. shaniehh

thank you