# Basic math study problems

### Problem 1

Simplify the following expressions as much as possible, without using a calculator. Do not approximate. Express all rational numbers as fractions. Show your work.

1. $$\frac{(2^3 \times 2^5)^{10}}{512}$$
2. $$(\log_2 13)(\log_{13} 2048)$$
3. $$\frac{\log_3(81^k)}{7k}$$
4. $$(1+i)(2-i)(3-i)$$, where $$i = \sqrt{-1}$$

Write out your solution, actually write it out on paper, before consulting the annotated solution