Question

If $a=3$ and $b=-2$, find the values of:
(i) $a^a+b^b$
(ii) $a^b+b^a$
(iii) ${\left(a+b\right)}^{ab}$

Answer 1

(i) $a^a+b^b$
Here, $a=3$ and $b=-2$.
Put the values in the expression $a^a+b^b$.
$$\begin{gathered} 3^3+{\left(-2\right)}^{-2} \\ =27+\frac{1}{{\left(-2\right)}^2} \\ =27+\frac{1}{4} \\ =\frac{108+1}{4} \\ =\frac{109}{4} \end{gathered}$$

(ii) $a^b+b^a$
Here, $a=3$ and $b=-2$.
Put the values in the expression $a^b+b^a$.
$$\begin{gathered} 3^{-2}+{\left(-2\right)}^3 \\ ={\left(\frac{1}{3}\right)}^2+\left(-8\right) \\ =\frac{1}{9}-8 \\ =\frac{1-72}{9} \\ =-\frac{71}{9} \end{gathered}$$

(iii) ${\left(a+b\right)}^{ab}$
Here, $a=3$ and $b=-2$.
Put the values in the expression ${\left(a+b\right)}^{ab}$.
$$\begin{gathered} {\left[3+\left(-2\right)\right]}^{3\left(-2\right)} \\ ={\left(1\right)}^{-6} \\ =1 \end{gathered}$$