3. A shopkeeper bought 300 apples at 80c each. 30 apples got rotten and the remaining were sold at the market for 20c each. Find the buying price.​

Sketch one cycle if the cosine function y=2cos2theta

1- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis.2- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in the first question?

Write the expression as a single natural logarithm. 2 in a - 4 in y