To solve the equation [tex]\(2^x + 2^{x+1} + 2^{x+2} + 2^{x+3} = 60\)[/tex], let's go through a detailed, step-by-step explanation:
1. Combine the terms with the same base:
We notice that each term on the left-hand side of the equation is a power of 2 where each exponent is increased by 1 in each subsequent term.
2. Factor out [tex]\(2^x\)[/tex] from the left-hand side:
[tex]\[
2^x + 2^{x+1} + 2^{x+2} + 2^{x+3} = 60
\][/tex]
This can be written as:
[tex]\[
2^x (1 + 2 + 2^2 + 2^3)
\][/tex]
3. Simplify inside the parentheses:
Calculate [tex]\(1 + 2 + 4 + 8\)[/tex]:
[tex]\[
1 + 2 + 4 + 8 = 15
\][/tex]
4. Rewrite the equation:
[tex]\[
2^x \cdot 15 = 60
\][/tex]
5. Isolate [tex]\(2^x\)[/tex]:
Divide both sides of the equation by 15:
[tex]\[
2^x = \frac{60}{15}
\][/tex]
Simplify the division:
[tex]\[
2^x = 4
\][/tex]
6. Express 4 as a power of 2:
Recall that [tex]\(4 = 2^2\)[/tex], so:
[tex]\[
2^x = 2^2
\][/tex]
7. Equate the exponents (since the bases are the same):
Therefore, we have:
[tex]\[
x = 2
\][/tex]
Considering the provided choices:
a. 00
b. 1
c. 2
d. 4
The correct answer is:
[tex]\[ \boxed{2} \][/tex]
