To solve the equation \((3^3)^x = 3^{24}\), we can use the property of exponents that states \((a^m)^n = a^{m \cdot n}\). Thus, we can rewrite the left side of the equation as follows:
\[ (3^3)^x = 3^{3x} \]
Now we have the equation:
\[ 3^{3x} = 3^{24} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 3x = 24 \]
Now, we can solve for \(x\) by dividing both sides by 3:
\[ x = \frac{24}{3} = 8 \]
Thus, the exponent that makes the two expressions equivalent is:
\[ \boxed{8} \]
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