To determine which expressions are equivalent to [tex]\(\frac{27^x}{9^x}\)[/tex], let's simplify the given expression step-by-step.
### Step-by-Step Simplification
1. Original Expression:
[tex]\[
\frac{27^x}{9^x}
\][/tex]
2. Rewrite as a Single Base:
Notice that both 27 and 9 can be expressed as powers of 3:
[tex]\[
27 = 3^3 \quad \text{and} \quad 9 = 3^2
\][/tex]
So, we can rewrite the expression using these bases:
[tex]\[
\frac{(3^3)^x}{(3^2)^x}
\][/tex]
3. Apply the Exponent Rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
Simplify the exponents:
[tex]\[
\frac{3^{3x}}{3^{2x}}
\][/tex]
4. Use the Quotient Rule for Exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
Simplify by subtracting the exponents:
[tex]\[
3^{3x - 2x} = 3^x
\][/tex]
### Equivalent Expressions Analysis
- Option A:
[tex]\[
3^x
\][/tex]
This is exactly what we obtained from the simplification. So, this is equivalent.
- Option B:
[tex]\[
(27-9)^x = 18^x
\][/tex]
This expression does not simplify to [tex]\(3^x\)[/tex]. So, it is not equivalent.
- Option C:
[tex]\[
9^x
\][/tex]
This is not equivalent to [tex]\(3^x\)[/tex].
- Option D:
[tex]\[
\left(\frac{27}{9}\right)^x = (3)^x = 3^x
\][/tex]
This is equivalent to [tex]\(3^x\)[/tex].
- Option E:
[tex]\[
\frac{9^x \cdot 3^x}{9^x} = \frac{9^x \cdot 3^x}{9^x} = 3^x
\][/tex]
Here, [tex]\(9^x\)[/tex] cancels out, leaving [tex]\(3^x\)[/tex]. So, this is equivalent.
- Option F:
This is simply the constant number 9, which isn't a function of [tex]\(x\)[/tex] and does not equate to [tex]\(3^x\)[/tex].
### Conclusion
The expressions equivalent to [tex]\(\frac{27^x}{9^x}\)[/tex] are:
[tex]\[
\boxed{\text{A, D, E}}
\][/tex]
