To solve the equation [tex]\((\sqrt{6})^{8x} = 216^{(x-3)}\)[/tex], let's go through the steps methodically:
1. Understand the Form of Each Side
We begin by expressing both sides of the equation in terms of their base numbers:
[tex]\[
(\sqrt{6})^{8x} \quad \text{and} \quad 216^{x-3}
\][/tex]
2. Simplify Bases
Recognize that [tex]\(\sqrt{6}\)[/tex] can be written as [tex]\(6^{1/2}\)[/tex], so the left-hand side becomes:
[tex]\[
(\sqrt{6})^{8x} = (6^{1/2})^{8x} = 6^{4x}
\][/tex]
Next, use the fact that [tex]\(216\)[/tex] can be factored into prime factors:
[tex]\[
216 = 6^3
\][/tex]
Therefore, the right-hand side becomes:
[tex]\[
216^{x-3} = (6^3)^{x-3} = 6^{3(x-3)} = 6^{3x-9}
\][/tex]
3. Equate Exponents
Since the bases are now the same, [tex]\(6\)[/tex], we can set the exponents equal to each other:
[tex]\[
4x = 3(x-3)
\][/tex]
4. Solve for [tex]\(x\)[/tex]
Now, solve the equation [tex]\(4x = 3(x-3)\)[/tex]:
[tex]\[
4x = 3x - 9
\][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
4x - 3x = -9
\][/tex]
Simplify:
[tex]\[
x = -9
\][/tex]
5. Verify Against Options
Given multiple choice options are:
- [tex]\(x = -9\)[/tex]
- [tex]\(x = -3\)[/tex]
- [tex]\(x = 0\)[/tex]
- [tex]\(x = 4\)[/tex]
The solution [tex]\(x = -9\)[/tex] is one of the given choices.
Therefore, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
