Alright, let's go through each given expression and check if it is equivalent to [tex]\(5^x\)[/tex].
### Expression A: [tex]\(\frac{15^x}{3^x}\)[/tex]
We can use properties of exponents to simplify this expression:
[tex]\[
\frac{15^x}{3^x} = \left(\frac{15}{3}\right)^x = 5^x
\][/tex]
Thus, Expression A is equivalent to [tex]\(5^x\)[/tex].
### Expression B: [tex]\(x^5\)[/tex]
This expression does not resemble or have any properties of exponents that can directly simplify to [tex]\(5^x\)[/tex]:
[tex]\[
x^5 \neq 5^x
\][/tex]
Thus, Expression B is not equivalent to [tex]\(5^x\)[/tex].
### Expression C: [tex]\(5 \cdot 5^{x+1}\)[/tex]
Let's simplify this expression using properties of exponents:
[tex]\[
5 \cdot 5^{x+1} = 5^1 \cdot 5^{x+1} = 5^{1 + x + 1} = 5^{x+2}
\][/tex]
Since [tex]\(5^{x+2} \neq 5^x\)[/tex], Expression C is not equivalent to [tex]\(5^x\)[/tex].
### Expression D: [tex]\(\frac{15^x}{3}\)[/tex]
This expression cannot be directly simplified to match [tex]\(5^x\)[/tex] because only the numerator is raised to the power [tex]\(x\)[/tex]:
[tex]\[
\frac{15^x}{3} \neq 5^x
\][/tex]
Thus, Expression D is not equivalent to [tex]\(5^x\)[/tex].
### Expression E: [tex]\(5 \cdot 5^{x-1}\)[/tex]
We simplify this expression by using properties of exponents:
[tex]\[
5 \cdot 5^{x-1} = 5^1 \cdot 5^{x-1} = 5^{1 + (x-1)} = 5^x
\][/tex]
Thus, Expression E is equivalent to [tex]\(5^x\)[/tex].
### Expression F: [tex]\(\left(\frac{15}{3}\right)^x\)[/tex]
We simplify this expression using properties of fractions and exponents:
[tex]\[
\left(\frac{15}{3}\right)^x = \left(5\right)^x = 5^x
\][/tex]
Thus, Expression F is equivalent to [tex]\(5^x\)[/tex].
So, the equivalent expressions to [tex]\(5^x\)[/tex] are:
- A. [tex]\(\frac{15^x}{3^x}\)[/tex]
- E. [tex]\(5 \cdot 5^{x-1}\)[/tex]
- F. [tex]\(\left(\frac{15}{3}\right)^x\)[/tex]
Therefore, the result is:
[tex]\[
\boxed{A, E, F}
\][/tex]
