To solve the equation [tex]\(8^{x+3} = 4^{x+1}\)[/tex], follow these steps:
1. Express [tex]\(8\)[/tex] and [tex]\(4\)[/tex] as powers of [tex]\(2\)[/tex]:
- [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex] because [tex]\(8 = 2 \times 2 \times 2 = 2^3\)[/tex].
- [tex]\(4\)[/tex] can be written as [tex]\(2^2\)[/tex] because [tex]\(4 = 2 \times 2 = 2^2\)[/tex].
2. Rewrite the given equation using these expressions:
[tex]\[
8^{x+3} = 4^{x+1}
\][/tex]
[tex]\[
(2^3)^{x+3} = (2^2)^{x+1}
\][/tex]
3. Simplify the expressions by using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(2^3)^{x+3} = 2^{3(x+3)} = 2^{3x + 9}
\][/tex]
[tex]\[
(2^2)^{x+1} = 2^{2(x+1)} = 2^{2x + 2}
\][/tex]
4. Now the equation becomes:
[tex]\[
2^{3x + 9} = 2^{2x + 2}
\][/tex]
5. Since the bases are the same (both are base [tex]\(2\)[/tex]), set the exponents equal to each other:
[tex]\[
3x + 9 = 2x + 2
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- Subtract [tex]\(2x\)[/tex] from both sides of the equation to isolate the terms involving [tex]\(x\)[/tex]:
[tex]\[
3x + 9 - 2x = 2x + 2 - 2x
\][/tex]
[tex]\[
x + 9 = 2
\][/tex]
- Subtract [tex]\(9\)[/tex] from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x + 9 - 9 = 2 - 9
\][/tex]
[tex]\[
x = -7
\][/tex]
Therefore, the solution to the equation [tex]\(8^{x+3} = 4^{x+1}\)[/tex] is [tex]\(\boxed{-7}\)[/tex].
