$\begin{aligned} 3^{27^3} & = 27^{3^x} \\ 3^{27^x} & = 3^{3 \cdot 3^x} \\ 27^x & = 3 \cdot 3^x \\ 3^{3x} & = 3^{x+1} \\ 3x & = x + 1 \\ \implies x & = \frac 12 = \boxed{0.5} \end{aligned}$

Know that $27=3^{3}$ And $3*3^{x}=3^{x+1}$

Now let's start re-writing these equations:P

$\large{3^{3^{3x}}=3^{3*3^{x}}}$

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$\large{3^{3^{3x}}=3^{3^{x+1}}}$

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$\large{=>3^{3x}=3^{x+1}}$

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$\large{3x=x+1}$

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$\boxed{\Huge{x=\dfrac{1}{2}}}$

Boom!

Solution Link

3^(27^x) = 27^(3^x)
= (3^3)^(3^x)
= 3^(3 × 3^x)
= 3^(3^(1 + x))

Same base, just taking the index now
27^x = (3^3)^x
= 3^(3x)
= 3^(1 + x)

Again, taking the index of the index with
3x = 1 + x
x = 0.5