1. What are ${\displaystyle 4^{3}}$, ${\displaystyle 3^{4}}$, ${\displaystyle 1^{250}}$, ${\displaystyle {250}^{1}?}$

a. ${\displaystyle \ 4^{3}=4\cdot 4\cdot 4=64}$
b. ${\displaystyle \ 3^{4}=3\cdot 3\cdot 3\cdot 3=81}$
c. ${\displaystyle \ 1^{250}=1\cdot 1\cdot 1\cdot \ .\,.\,.\,.\ \cdot 1\cdot 1\cdot 1=1}$
d. ${\displaystyle \ {250}^{1}=250}$

2. Write these numbers as powers of 2: ${\displaystyle 128,8,1024}$

a. ${\displaystyle \ 128=2^{7}}$
b. ${\displaystyle \ 8=2^{3}}$
c. ${\displaystyle \ 1024=2^{10}}$

3. What is ${\displaystyle (2^{3})*(2^{2})?}$

${\displaystyle (2^{3})\cdot (2^{2})=2^{3+2}=2^{5}=32}$

4. What is ${\displaystyle (2^{6})/(2^{2})?}$

${\displaystyle (2^{6})/(2^{2})=2^{6-2}=2^{4}=16\ }$

5. Harder: Why does ${\displaystyle 3^{0}=1}$? (clue: think about ${\displaystyle {3^{2}}/{3^{2}}}$, for example)

${\displaystyle {3^{2}}/{3^{2}}=3^{2-2}=3^{0}\ }$
${\displaystyle 3^{2}=3\cdot 3=9}$
${\displaystyle {3^{2}}/{3^{2}}=9/9=1\ }$
From the first and third equations above, we can see that:
${\displaystyle {3^{2}}/{3^{2}}=3^{0}=1\ }$
The exponent doesn't have to be 2. The exponent can be any real number and the same logic would work. It was mentioned in the first Arithmetic chapter that a number raised to the 0 power equals 1; i. e.,
${\displaystyle a^{0}=1\ }$ .