To solve the equation [tex]\(8^{3^{x+1}} = 2^{9^{x-1}}\)[/tex], we can follow these steps:
1. Express Both Sides with the Same Base:
- Notice that 8 can be expressed as [tex]\(2^3\)[/tex] and 9 can be expressed as [tex]\(3^2\)[/tex]. Let's rewrite both sides accordingly.
[tex]\[
8^{3^{x+1}} = (2^3)^{3^{x+1}}
\][/tex]
and
[tex]\[
2^{9^{x-1}} = 2^{(3^2)^{x-1}}
\][/tex]
2. Simplify the Exponents:
- Apply the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
[tex]\[
(2^3)^{3^{x+1}} = 2^{3 \cdot 3^{x+1}}
\][/tex]
and
[tex]\[
2^{(3^2)^{x-1}} = 2^{(3^{2(x-1)})}
\][/tex]
Simplify inside the exponent on the right-hand side:
[tex]\[
2^{(3^{2(x-1)})} = 2^{(3^{2x-2})}
\][/tex]
3. Set Exponents Equal to Each Other:
- Since the bases are the same (both base 2), we can set the exponents equal to each other:
[tex]\[
3 \cdot 3^{x+1} = 3^{2x-2}
\][/tex]
4. Simplify the Equation:
- Recognize that [tex]\(3^{x+1} = 3 \cdot 3^x\)[/tex]. Replace this back into the equation:
[tex]\[
3 \cdot 3 \cdot 3^x = 3^{2x-2}
\][/tex]
Simplify the left-hand side:
[tex]\[
3^2 \cdot 3^x = 3^{2x-2}
\][/tex]
[tex]\[
3^{2 + x} = 3^{2x-2}
\][/tex]
5. Equate the Exponents:
- Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2 + x = 2x - 2
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
2 + x = 2x - 2
\][/tex]
[tex]\[
2 + 2 = 2x - x
\][/tex]
[tex]\[
4 = x
\][/tex]
So, the solution to the equation [tex]\(8^{3^{x+1}} = 2^{9^{x-1}}\)[/tex] is [tex]\(x = 4\)[/tex].
