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

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

2. Write these numbers as powers of 2: $128, 8, 1024$

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

3. What is $(2^3)*(2^2)?$

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

4. What is $(2^6)/(2^2)?$

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

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

${3^2}/{3^2} = 3^{2-2} = 3^0 \$
$3^2 = 3 \cdot 3 = 9$
${3^2}/{3^2} = 9/9 = 1 \$
From the first and third equations above, we can see that:
${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.,
$a^0 = 1 \$ .