Why is the product of a rational number and an irrational number irrational?

Step-by-step explanation:

When you multiply and irrational by a rational, you are doing nothing to change the nature of an irrational number. The two numbers have different kinds of properties and though a multiplication can be indicated, there is no way you can change the properties of an irrational number with a rational.

Their properties just do not mix.

It is like take a solvent and mixing it with something that is not a solvent. You cannot do anything to anything to the insolvent to make it solvent simply by adding more. Their properties remain in tact.

Take 3 * 4√7

The properties of the 3 and 4 remain the same when multiplied and they can be combined. 3*4 = 12

But nothing can be done with √7. The 3 cannot effect it, anymore than what the 4 did.

i can you with this question sweetie!

step-by-step explanation:

you didn't post any answer choices, so its not 100% clear what your teacher is looking for

however, log(15/7) is equivalent to log(15) - log(7)

i'm using the rule log(x/y) = log(x) - log(y)