To solve the equation
[tex]\[
\sqrt[3]{9 + \sqrt{x}} + \sqrt[3]{9 - \sqrt{x}} = 3,
\][/tex]
we start by letting
[tex]\( a = \sqrt[3]{9 + \sqrt{x}} \)[/tex]
and
[tex]\( b = \sqrt[3]{9 - \sqrt{x}} \)[/tex].
This allows us to rewrite the equation as:
[tex]\[
a + b = 3.
\][/tex]
Next, we cube both sides of the equation to eliminate the cube roots, which gives us:
[tex]\[
(a + b)^3 = 3^3.
\][/tex]
Expanding the left-hand side, we get:
[tex]\[
a^3 + b^3 + 3ab(a + b) = 27.
\][/tex]
Since [tex]\(a + b = 3\)[/tex], we can substitute [tex]\(a + b\)[/tex] into the equation:
[tex]\[
a^3 + b^3 + 3ab \cdot 3 = 27.
\][/tex]
This simplifies to:
[tex]\[
a^3 + b^3 + 9ab = 27.
\][/tex]
Next, we recognize that [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are cube roots of the expressions [tex]\(9 + \sqrt{x}\)[/tex] and [tex]\(9 - \sqrt{x}\)[/tex], respectively. Thus,
[tex]\[
a^3 = 9 + \sqrt{x} \quad \text{and} \quad b^3 = 9 - \sqrt{x}.
\][/tex]
So,
[tex]\[
a^3 + b^3 = (9 + \sqrt{x}) + (9 - \sqrt{x}) = 18.
\][/tex]
Substituting this back into our equation, we get:
[tex]\[
18 + 9ab = 27.
\][/tex]
Subtracting 18 from both sides, we have:
[tex]\[
9ab = 9.
\][/tex]
So,
[tex]\[
ab = 1.
\][/tex]
Now, let's recall that [tex]\(a\)[/tex] and [tex]\(b\)[/tex] were defined as:
[tex]\[
a = \sqrt[3]{9 + \sqrt{x}} \quad \text{and} \quad b = \sqrt[3]{9 - \sqrt{x}}.
\][/tex]
So,
[tex]\[
ab = \sqrt[3]{(9 + \sqrt{x})(9 - \sqrt{x})}.
\][/tex]
We know [tex]\(ab = 1\)[/tex], so:
[tex]\[
\sqrt[3]{(9 + \sqrt{x})(9 - \sqrt{x})} = 1.
\][/tex]
Cubing both sides to eliminate the cube root, we get:
[tex]\[
(9 + \sqrt{x})(9 - \sqrt{x}) = 1.
\][/tex]
Expanding the left-hand side using the difference of squares, we have:
[tex]\[
81 - (\sqrt{x})^2 = 1.
\][/tex]
Simplifying this, we obtain:
[tex]\[
81 - x = 1.
\][/tex]
Solving for [tex]\(x\)[/tex], we subtract 1 from both sides:
[tex]\[
81 - 1 = x.
\][/tex]
Thus,
[tex]\[
x = 80.
\][/tex]
Therefore, the solution is:
[tex]\[
\boxed{80}
\][/tex]
