1. The value of the real number $a$ that satisfies the equation $\sqrt{1+\sqrt{1+a}}=\sqrt[3]{a}$ is $\qquad$ .
2, 1.8.
From the given equation, we have
$$
(1+\sqrt{1+a})^{3}=a^{2}(a>1) \text {. }
$$

Let $x=\sqrt{1+a}$. Then $x \geqslant 0$, and $a=x^{2}-1$. Thus, $(1+x)^{3}=\left(x^{2}-1\right)^{2}$.
After rearranging and factoring, we get
$$
\begin{array}{l}
x(x-3)(x+1)^{2}=0 \\
\Rightarrow x_{1}=0, x_{2}=3, x_{3}=-1 \text { (discard) } \\
\Rightarrow a=-1 \text { (discard) or } a=8 .
\end{array}
$$