If a = 3 and b = -2, find the values of:

(i) a^a + b^b

(ii) a^b + b^a

(i) Given, a = 3 and b = -2.

Substituting values of a and b in a^a + b^b we get,

= 3³ + (-2)^-2

= 27 + $\Big(-\dfrac{1}{2}\Big)^2$

= 27 + $\dfrac{1}{4} = 27\dfrac{1}{4}$.

Hence, $a^a + b^b = 27\dfrac{1}{4}.$

(ii) Given, a = 3 and b = -2.

Substituting values of a and b in a^b + b^a we get,

$= (3)^{-2} + (-2)^3 \\[1em] = \Big(\dfrac{1}{3}\Big)^2 + (-8) \\[1em] = \dfrac{1}{9} - 8 \\[1em] = \dfrac{1 - 72}{9} \\[1em] = -\dfrac{71}{9} = -7\dfrac{8}{9}.$

Hence, $a^a + b^b = -7\dfrac{8}{9}.$