Simple

We simply square the expression: $(a + \frac{1}{a})^2 = 5^2 \\ \implies a^2 + 2 + \frac{1}{a^2} = 25 \\ \implies a^2 + \frac{1}{a^2} = 23$

simple square both sides and subtract 2 from the value... we get 25-2 = 23

a^2+(1/a)^2=(a+1/a)^2-2 a 1/a=(5)^2-2=25-2=23

Wouldn't a better solution be squaring a + 1/a=5 then deriving at a^2+2+1/a^2=25? Hence, the answer would be 23 as a^2+1/a^2=25-2.

(a + 1/a )^2 = 25

(a^2) + 2 + (1/a)^2 = 25

(a^2) + (1/a)^2 = 23

Note that

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

Hence,

$a^2+\frac{1}{a^2}=25-2=\boxed{23},$

and we are done.

I just ended up with trial and error, ending up with $a$ as $4.75$ .

$4.75 + \frac{1}{4.75} = 4.96052631578...$ which estimates to $5$

$4.75^{2} = 22.5625$

$22.5625 + \frac{1}{22.5625} = 22.6068213296...$ which estimates to $\boxed{23}$