(Exponential Functions and Function Combinations MC)

Which of the following expressions is equivalent to [tex]2^{3x-4}[/tex]?

A. [tex]\frac{1^x}{8}[/tex]
B. [tex]\frac{3^x}{4}[/tex]
C. [tex]\frac{6^x}{8}[/tex]
D. [tex]\frac{8^x}{16}[/tex]



Answer :

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]