Let's solve the equation [tex]\(2^{x+3} + 2^x = 36\)[/tex] step-by-step.
1. Express the equation in terms of [tex]\(2^x\)[/tex]:
Notice that [tex]\(2^{x+3}\)[/tex] can be rewritten using the properties of exponents. Specifically, [tex]\(2^{x+3} = 2^x \cdot 2^3 = 8 \cdot 2^x\)[/tex].
Therefore, the given equation [tex]\(2^{x+3} + 2^x = 36\)[/tex] can be rewritten as:
[tex]\[
8 \cdot 2^x + 2^x = 36
\][/tex]
2. Combine like terms:
Factor out [tex]\(2^x\)[/tex] from both terms on the left side of the equation:
[tex]\[
8 \cdot 2^x + 2^x = (8 + 1) \cdot 2^x = 9 \cdot 2^x
\][/tex]
Hence, the equation simplifies to:
[tex]\[
9 \cdot 2^x = 36
\][/tex]
3. Isolate [tex]\(2^x\)[/tex]:
Solve for [tex]\(2^x\)[/tex] by dividing both sides of the equation by 9:
[tex]\[
2^x = \frac{36}{9} = 4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
We know that [tex]\(4\)[/tex] can be expressed as a power of 2. Specifically, [tex]\(4 = 2^2\)[/tex].
Therefore, we have:
[tex]\[
2^x = 2^2
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
x = 2
\][/tex]
Thus, the solution to the equation [tex]\(2^{x+3} + 2^x = 36\)[/tex] is:
[tex]\[
x = 2
\][/tex]
