Given,

$\rm 2^{x+3} + 2^x = 36$

By using the law of indices, $\rm a^{m+n} = a^m \cdot a^n$, we get,

$\rm or, 2^x \cdot 2^3 + 2^x = 36$

$\rm or, 2^x \cdot ( 2^3 + 1) = 36$

$\rm or, 2^x \cdot (8 + 1) = 36$

$\rm or, 2^x \cdot 9 = 36$

Dividing both sides of the equation by 9, we get,

$\rm or, 2^x \cdot \frac{9}{9} = \frac{36}{9}$

$\rm or, 2^x = 4$

$\rm or, 2^x = 2^2$

The bases of the terms on both sides of the equation are the same, so we equate their powers.

$\rm \therefore x = 2$

Hence, the required value of x is 2.