Let's analyze each expression and determine whether it is equivalent to [tex]\( 5^x \)[/tex].
### Expression [tex]\( A \)[/tex]: [tex]\( 5 \cdot 5^{x-1} \)[/tex]
Using the properties of exponents, we can simplify this expression:
[tex]\[ 5 \cdot 5^{x-1} = 5^1 \cdot 5^{x-1} = 5^{1 + (x-1)} = 5^x \][/tex]
Thus, [tex]\( 5 \cdot 5^{x-1} \)[/tex] is indeed equivalent to [tex]\( 5^x \)[/tex].
### Expression [tex]\( B \)[/tex]: [tex]\( x^5 \)[/tex]
This expression does not match [tex]\( 5^x \)[/tex]. The base and exponent are switched, so:
[tex]\[ x^5 \neq 5^x \][/tex]
Thus, [tex]\( x^5 \)[/tex] is not equivalent to [tex]\( 5^x \)[/tex].
### Expression [tex]\( C \)[/tex]: [tex]\( \frac{15^x}{3^x} \)[/tex]
We can use the property of exponents where [tex]\( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)[/tex]:
[tex]\[ \frac{15^x}{3^x} = \left(\frac{15}{3}\right)^x = 5^x \][/tex]
Thus, [tex]\( \frac{15^x}{3^x} \)[/tex] is equivalent to [tex]\( 5^x \)[/tex].
### Expression [tex]\( D \)[/tex]: [tex]\( \frac{15^x}{3} \)[/tex]
Here, we divide [tex]\( 15^x \)[/tex] by [tex]\( 3 \)[/tex], which does not simplify to [tex]\( 5^x \)[/tex]:
[tex]\[ \frac{15^x}{3} \neq 5^x \][/tex]
Thus, [tex]\( \frac{15^x}{3} \)[/tex] is not equivalent to [tex]\( 5^x \)[/tex].
### Expression [tex]\( E \)[/tex]: [tex]\( \left(\frac{15}{3}\right)^x \)[/tex]
Since [tex]\( \frac{15}{3} = 5 \)[/tex], we have:
[tex]\[ \left(\frac{15}{3}\right)^x = 5^x \][/tex]
Thus, [tex]\( \left(\frac{15}{3}\right)^x \)[/tex] is equivalent to [tex]\( 5^x \)[/tex].
### Expression [tex]\( F \)[/tex]: [tex]\( 5 \cdot 5^{x+1} \)[/tex]
Using the properties of exponents, we can simplify this expression:
[tex]\[ 5 \cdot 5^{x+1} = 5^1 \cdot 5^{x+1} = 5^{1 + (x+1)} = 5^{x+2} \][/tex]
Thus, [tex]\( 5 \cdot 5^{x+1} \)[/tex] is not equivalent to [tex]\( 5^x \)[/tex].
### Summary
The expressions that are equivalent to [tex]\( 5^x \)[/tex] are:
- [tex]\( A: 5 \cdot 5^{x-1} \)[/tex]
- [tex]\( C: \frac{15^x}{3^x} \)[/tex]
- [tex]\( E: \left(\frac{15}{3}\right)^x \)[/tex]
