First, we need to simplify each of the expression using rules for exponents and then evaluate with z = 12:
color(red)(A)) Use these rules for exponents:
x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) +color(blue)(b))
x^color(red)(a) = 1/x^color(red)(-a)
z^-6z^4 = z^(-6+4) = z^-2 = 1/z^(- -2) = 1/z^2
1/(12)^2 = 1/144 = 0.00694
color(red)(B)) Use these rules for exponents:
x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) +color(blue)(b))
x^color(red)(a) = 1/x^color(red)(-a)
(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))
(z^-2z^5)^-2 = (z^(-2+5))^-2 = (z^3)^-2 = z^(3xx-2) = z^-6 = 1/z^6
1/(12)^6 = 1/2985984 = 3.35 x 10^-7 = 0.000000335
color(red)(C)) Use these rules for exponents:
(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))
(z^3)^5 = z^(3xx5) = z^15
12^15 = 1.54 x 10^16 = 15,400,000,000,000,000
color(red)(D)) Use these rules for exponents:
x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) +color(blue)(b))
(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))
-(z^2z^-4)^-3 = -(z^(2-4))^-3 = -(z^-2)^-3 = -z^(-2xx-3) = -z^6
-(12)^6 = -2,985,984
