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