To solve for the exponent \( x \) in the equation \( (16^x)^x = (16^{18}) \), we start by simplifying the left side of the equation.
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can rewrite \( (16^x)^x \) as follows:
\[ (16^x)^x = 16^{x \cdot x} = 16^{x^2} \]
Thus, the equation becomes:
\[ 16^{x^2} = 16^{18} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x^2 = 18 \]
Now, we solve for \( x \) by taking the square root of both sides:
\[ x = \pm \sqrt{18} \]
We can simplify \( \sqrt{18} \):
\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]
Thus, the solutions for \( x \) are:
\[ x = 3\sqrt{2} \quad \text{or} \quad x = -3\sqrt{2} \]
Therefore, the exponents that would make the two expressions equivalent are:
\[ \boxed{\pm 3\sqrt{2}} \]