Let's simplify the given expression step-by-step to find which expressions are equivalent.
The given expression is:
[tex]\[
\frac{16^x}{4^x}
\][/tex]
First, rewrite [tex]\(16^x\)[/tex] in terms of a base that is related to 4. We know that [tex]\(16\)[/tex] is [tex]\(4^2\)[/tex], so:
[tex]\[
16^x = (4^2)^x = 4^{2x}
\][/tex]
Substituting [tex]\(16^x\)[/tex] with [tex]\(4^{2x}\)[/tex], we get:
[tex]\[
\frac{4^{2x}}{4^x}
\][/tex]
Now, use the properties of exponents. Specifically, [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{4^{2x}}{4^x} = 4^{2x - x} = 4^x
\][/tex]
So, we have simplified the expression [tex]\(\frac{16^x}{4^x}\)[/tex] to [tex]\(4^x\)[/tex].
Now let's evaluate each of the given options to check their equivalence to [tex]\(4^x\)[/tex]:
- A. [tex]\(4^x\)[/tex]:
This is exactly the simplified form. This is correct.
- B. [tex]\(4\)[/tex]:
This is not equivalent to [tex]\(4^x\)[/tex] because [tex]\(4\)[/tex] is a constant while [tex]\(4^x\)[/tex] changes with [tex]\(x\)[/tex]. This is incorrect.
- C. [tex]\(\left(\frac{16}{4}\right)^x\)[/tex]:
We can simplify [tex]\(\frac{16}{4}\)[/tex]:
[tex]\[
\frac{16}{4} = 4
\][/tex]
So:
[tex]\[
\left(\frac{16}{4}\right)^x = 4^x
\][/tex]
This is correct.
- D. [tex]\(16^x\)[/tex]:
Comparing this with our simplified expression, this was part of the original expression, not the simplified one. This is incorrect.
- E. [tex]\((16-4)^x\)[/tex]:
Simplify inside the parenthesis:
[tex]\[
16 - 4 = 12
\][/tex]
So:
[tex]\[
(16-4)^x = 12^x
\][/tex]
This is not equivalent to [tex]\(4^x\)[/tex]. This is incorrect.
- F. [tex]\(\frac{4^x=4^x}{4^x}\)[/tex]:
The expression within the fraction is confusing and seems to contain a typographical error. Assuming it is meant to be [tex]\(\frac{4^x}{4^x}\)[/tex], this simplifies to:
[tex]\[
\frac{4^x}{4^x} = 1
\][/tex]
This is not equivalent to [tex]\(4^x\)[/tex]. This is incorrect.
So the correct options are:
[tex]\[
\text{A. } 4^x \text{ and C. } \left(\frac{16}{4}\right)^x
\][/tex]
