To solve the problem, we need to determine which of the given expressions is equivalent to [tex]\(2^{3x-4}\)[/tex]. Let's investigate each option one by one.
Option 1: [tex]\(\frac{1^x}{8}\)[/tex]
We know that [tex]\(1^x = 1\)[/tex] for any value of [tex]\(x\)[/tex], so:
[tex]\[
\frac{1^x}{8} = \frac{1}{8}
\][/tex]
This is a constant value and does not depend on [tex]\(x\)[/tex]. Clearly, this does not match the form of [tex]\(2^{3x-4}\)[/tex], which depends on [tex]\(x\)[/tex].
Option 2: [tex]\(\frac{3^x}{4}\)[/tex]
This expression is already in its simplest form:
[tex]\[
\frac{3^x}{4}
\][/tex]
This is different from [tex]\(2^{3x-4}\)[/tex] since the base of the exponent (3) is not the same and the form does not match.
Option 3: [tex]\(\frac{6^x}{8}\)[/tex]
This expression can be analyzed by considering its components:
[tex]\[
\frac{6^x}{8}
\][/tex]
which does not simplify in any way to match [tex]\(2^{3x-4}\)[/tex] because [tex]\(6\)[/tex] and [tex]\(8\)[/tex] are not powers of [tex]\(2\)[/tex] in such a way that we could simplify it to [tex]\(2^{3x-4}\)[/tex].
Option 4: [tex]\(\frac{8^x}{16}\)[/tex]
First, recognize that [tex]\(8\)[/tex] can be written as:
[tex]\[
8 = 2^3 \quad \text{so} \quad 8^x = (2^3)^x = 2^{3x}
\][/tex]
Now, rewrite the given expression [tex]\( \frac{8^x}{16} \)[/tex] using the fact that [tex]\(8^x = 2^{3x}\)[/tex]:
[tex]\[
\frac{8^x}{16} = \frac{2^{3x}}{16}
\][/tex]
Next, express [tex]\(16\)[/tex] as a power of 2:
[tex]\[
16 = 2^4
\][/tex]
Thus, we have:
[tex]\[
\frac{2^{3x}}{16} = \frac{2^{3x}}{2^4}
\][/tex]
When dividing by a power of the same base, subtract the exponents:
[tex]\[
\frac{2^{3x}}{2^4} = 2^{3x - 4}
\][/tex]
This matches exactly with [tex]\(2^{3x-4}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{8^x}{16}}
\][/tex]
