# The Art of Computer Programming/Numbers, Powers, and Logarithms/Exercises

My (Bakert 20:26, 15 October 2006 (UTC)) notes on these exercises:

1. I thought 1/infinity but that seemed like cheating. The real answer is that there are none - whatever number you think of, it is bigger than the number divided by 2.

2. I said yes but with infinitely many 9s the answer is apparently no, because the decimal expansion would be 0.24.

Knuth is rather sparse on the meaning of decimal expansion. - Luckly there is a pretty good entry in Wikipedia http://en.wikipedia.org/wiki/Decimal_expansion

3. ${\displaystyle (-3)^{-3}={\frac {1}{(-3)^{3}}}={\frac {1}{-27}}}$

4. To raise something to the 1/3 power means to take the cube root. So to raise something to the 2/3 power means take the cube root of the number and square it. That the power is negative means we need 1/result. So I thought this meant we needed:

${\displaystyle {\frac {1}{{\sqrt[{3}]{0.125}}^{2}}}={\frac {1}{.5^{2}}}={\frac {1}{0.25}}=4}$