Select from the drop-down menus to correctly identify whether the given operation is closed or not closed with respect to each set of numbers.

real numbers are (closed, not closed) under multiplication.

irrational numbers are (closed, not closed) under division.

integers are (closed, not closed) under subtraction.

positive rational numbers are (closed, not closed) under addition.

An automated water dispenser fills packets with one liter of water on average, with a standard deviation of 5 milliliter. the manual says that after a year of operation the dispenser should be tested to see if it needs recalibration. a year later a number of filled packets are set aside and measured separately. it is found that the average packet now contains about 0.995 liters. does the dispenser need calibration? explain your answer.

If cos(2x) = tan^2(y), show that cos(2y) = tan^2(x).

Simplify. 3(4+4x) (type your answer in with no spaces)