Let's begin by simplifying the given expression:
[tex]\[
\frac{21^x}{7^x}
\][/tex]
Step 1: Apply the rules of exponents. The expression can be rewritten by combining the exponents over a single fraction:
[tex]\[
\frac{21^x}{7^x} = \left(\frac{21}{7}\right)^x
\][/tex]
Step 2: Simplify the fraction inside the parentheses:
[tex]\[
\left(\frac{21}{7}\right)^x = 3^x
\][/tex]
So, the expression [tex]\(\frac{21^x}{7^x}\)[/tex] simplifies to [tex]\(3^x\)[/tex].
Now let's check each given expression to see if they are equivalent to [tex]\(3^x\)[/tex]:
A. [tex]\(\frac{7^x \cdot 3^x}{7^x}\)[/tex]
Simplification:
[tex]\[
\frac{7^x \cdot 3^x}{7^x} = \frac{7^x}{7^x} \cdot 3^x = 1 \cdot 3^x = 3^x
\][/tex]
Therefore, this is equivalent to [tex]\(3^x\)[/tex].
B. [tex]\(\left(\frac{21}{7}\right)^x\)[/tex]
Simplification:
[tex]\[
\left(\frac{21}{7}\right)^x = 3^x
\][/tex]
Therefore, this is equivalent to [tex]\(3^x\)[/tex].
C. [tex]\(3^{x-7}\)[/tex]
Simplification:
[tex]\[
3^{x-7} \neq 3^x \quad \text{(not equivalent)}
\][/tex]
D. [tex]\(3\)[/tex]
Simplification:
[tex]\[
3 \neq 3^x \quad \text{(not equivalent unless } x = 1 \text{, but this is not generally true for all x)}
\][/tex]
E. [tex]\(3^x\)[/tex]
This is already in the required form [tex]\(3^x\)[/tex].
F. [tex]\((21-7)^x\)[/tex]
Simplification:
[tex]\[
(21-7)^x = 14^x \neq 3^x \quad \text{(not equivalent)}
\][/tex]
The expressions that are equivalent to [tex]\(\frac{21^x}{7^x}\)[/tex] (which simplifies to [tex]\(3^x\)[/tex]) are:
- A. [tex]\(\frac{7^x \cdot 3^x}{7^x}\)[/tex]
- B. [tex]\(\left(\frac{21}{7}\right)^x\)[/tex]
- E. [tex]\(3^x\)[/tex]
