Let's solve the given equation step by step:
We have:
[tex]\[ 2^{x+3} + 2^x = 36 \][/tex]
First, let's rewrite [tex]\( 2^{x+3} \)[/tex] using properties of exponents:
[tex]\[ 2^{x+3} = 2^x \cdot 2^3 = 2^x \cdot 8 \][/tex]
Now, substitute this back into the original equation:
[tex]\[ 8 \cdot 2^x + 2^x = 36 \][/tex]
Notice that both terms on the left-hand side have a common factor of [tex]\( 2^x \)[/tex]. So, we can factor out [tex]\( 2^x \)[/tex]:
[tex]\[ 2^x (8 + 1) = 36 \][/tex]
Simplify inside the parentheses:
[tex]\[ 2^x \cdot 9 = 36 \][/tex]
Now, divide both sides by 9 to solve for [tex]\( 2^x \)[/tex]:
[tex]\[ 2^x = \frac{36}{9} \][/tex]
Simplify the right-hand side:
[tex]\[ 2^x = 4 \][/tex]
We know that [tex]\( 4 \)[/tex] can be expressed as a power of 2:
[tex]\[ 4 = 2^2 \][/tex]
So:
[tex]\[ 2^x = 2^2 \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ x = 2 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
Thus the correct answer is:
a. 2
