Exercise 5:
You need to use these rules:
1) [tex] (ab)^c = a^c\cdot b^c [/tex] (the exponent distributes on factors)
2) [tex] (a^b)^c = a^{bc} [/tex] (the exponents multiplies in case of multiple exponentiations)
So, you have
[tex] (5a^2)^3 \xrightarrow{\text{Rule 1}} 5^3 (a^2)^3 \xrightarrow{\text{Rule 2}} 5^3 a^6 [/tex]
Exercise 6-15-18:
Exactly the same as exercise 5
Exercise 11:
You have to use the rule [tex] a^b \cdot a^c = a^{b+c} [/tex] (let's call it Rule 3). In other words, if you multiply two powers of the same base, you have a power of that same base, whose exponent is the sum of the exponents.
So, you have
[tex] 4^7 \cdot 3^2 \cdot 3^4 \cdot 4^{10} \xrightarrow{\text{Commutative}} 4^7\cdot 4^{10} \cdot 3^2 \cdot 3^4 \xrightarrow{\text{Rule 3}} 4^{7+10} \cdot 3^{2+4} = 4^{17} \cdot 3^6 [/tex]
