Certainly! Let's solve the equation [tex]\(2^{x+1} - 2^x = 8\)[/tex] step by step.
1. Start with the given equation:
[tex]\[
2^{x+1} - 2^x = 8
\][/tex]
2. Recognize that [tex]\(2^{x+1}\)[/tex] can be rewritten using the properties of exponents:
[tex]\[
2^{x+1} = 2 \cdot 2^x
\][/tex]
3. Substitute [tex]\(2 \cdot 2^x\)[/tex] for [tex]\(2^{x+1}\)[/tex] in the equation:
[tex]\[
2 \cdot 2^x - 2^x = 8
\][/tex]
4. Factor out [tex]\(2^x\)[/tex] from the left side of the equation:
[tex]\[
2^x (2 - 1) = 8
\][/tex]
5. Simplify the equation:
[tex]\[
2^x \cdot 1 = 8 \implies 2^x = 8
\][/tex]
6. Recognize that [tex]\(8\)[/tex] can be written as a power of [tex]\(2\)[/tex]:
[tex]\[
8 = 2^3
\][/tex]
7. Set the exponents equal to each other since the bases are the same:
[tex]\[
x = 3
\][/tex]
Therefore, the solution to the equation [tex]\(2^{x+1} - 2^x = 8\)[/tex] is:
[tex]\[
x = 3
\][/tex]
