Math Is Fun Forum

The value

The problem states that there is a shortcut and big calculations are unnecessary and probably part of a wrong solution.
I have found that the complex number without the power of 72 multiplied by its conjugate is the its magnitude squared. So, the magnitude must be a real number. However, I don't know what to do with this information.

In the diagram below, we have $\angle ABC = \angle ACB = \angle DEC=\angle CDE$, $BC = 8$, and $DB = 2$. Find $AB$.

Angle bisectors $\overline{TX}$ and $\overline{UY}$ of $\triangle TUV$ meet at point $I$. Find all possible values of $\angle V$ if $\angle  TIU = 109^\circ$. As your answer, enter the number of degrees in $\angle V$. If you find more than one possibility, list the possible values in increasing order, separated by commas.

The interior angle measures of a pentagon form an arithmetic progression. The difference between the largest and smallest angle measures is $44^\circ$. Find the measure of the smallest angle, in degrees.

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, $S$, $T$, $U$, $V$, and $W$ be the trisection points of the sides of $ABCD$, as shown.
http://latex.artofproblemsolving.com/8/3/d/83df21d419d8743fb5b781faac136a6c8676d4ef.png
If the area of quadrilateral $ABCD$ is 180, then find the area of hexagon $AQRCUV$.

In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there?

Help very greatly appreciated!

Alana is making a ball out of rubber bands. When the ball has 54 rubber bands, it has a diameter of 3 cm. How many rubber bands should Alana add to the ball to increase its diameter by 1 cm? Assume that all of Alana's rubber bands have the same volume.

I am getting 74. Could anyone please tell me if that's correct? Thanks.