Alan

As follows:

The prime factors of 180 are 2². 3². 5  Hence to make a cube we need to multiply by 2.3.5² = 150

180*150 = 27000

27000^1/3 = 30

"Elmer has taken n tests.  If he receives 73  on the next test, his average will be exactly 83.  If he receives 99, then the average will be exactly 87.  What does n equal?"

(x + 73)/(n + 1) = 83   ...(1)

(x + 99)/(n + 1) = 87   ...(2)

Rewrite (1) as x + 73 = 83*(n + 1)   ...(3)

and (2) as       x + 99 = 87*(n + 1)   ...(4)

Subtract (3) from (4):  26 = 4n + 4  so  n = 5.5    Elmer took 5.5 tests??????

All of them!

This is (part of) the Collatz conjecture.

As follows:

Here's a more detailed solution that agrees with Geno, that it isn't possible to do it within the specified cost:

To find the product, multiply the magnitudes and add the angles.

As follows:

Of course, it helps to know the summations!  They occur often enough that they are worth memorising.

A complex number in the form: z = a + bi  can be written in polar form as: z = r.e^i.theta, where r = √(a² + b²) and theta = tan^-1(b/a)  (that's arctan(b/a))

You should plug your numbers into these.

I notice the polar form sometimes seems to be written as  r.cis(theta)  which is r.cos(theta) + i.r.sin(theta)  (which I think is more like rectangular form!!).

Let n = 99999 - 10000 + 1 = 90000

n/3 + n/5 - n/15 = 42000    (we subtract n/15 to avoid double counting those that are divisible by both 3 and 5).