If a=3 and b=−2, find the values of:

(i) $a^{a}+b^{b}$

Here, $a=3$ and $b=-2$.

Put the values in the expression $a^{a}+b^{b}$.

$3^{3}+(-2)^{-2}$

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

$=27+\frac{1}{4}$

$=\frac{108+1}{4}$

$=\frac{109}{4}$

(ii) $a^{b}+b^{a}$

Here, $a=3$ and $b=-2$

Put the values in the expression $a^{b}+b^{a}$.

$3^{-2}+(-2)^{3}$

$=\left(\frac{1}{3}\right)^{2}+(-8)$

$=\frac{1}{9}-8$

$=\frac{1-72}{9}$

$=-\frac{71}{9}$

(iii) $(a+b)^{a b}$

Here, $a=3$ and $b=-2$

Put the values in the expression $(a+b)^{a b}$.

$[3+(-2)]^{3(-2)}$

$=(1)^{-6}$

$=1$