# 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.

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

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