114. Calculate the sum:

$$
x=\sqrt[3]{9+4 \sqrt{5}}+\sqrt[3]{9-4 \sqrt{5}}
$$
$$
\triangle$ Let's raise the equality to the third power, using the formula

$$
(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)
$$

We get:

$$
\begin{aligned}
& x^{3}=(9+4 \sqrt{5})+(9-4 \sqrt{5})+3 \sqrt[3]{81-80} \cdot(\sqrt[3]{9+4 \sqrt{5}}+\sqrt[3]{9-4 \sqrt{5}}) \\
& x^{3}=18+3 x, \quad x^{3}-3 x-18=0
\end{aligned}
$$

The last equation has an integer root $x=3$. It can be rewritten as

$$
(x-3)\left(x^{2}+3 x+6\right)=0
$$

The equation has no other real roots.

Answer: 3.
$$