Answer :
Let's determine which expressions are equivalent to [tex]\( 3^x \)[/tex].
### Option A: [tex]\( 3 \cdot 3^{x-1} \)[/tex]
We can use the rules of exponents to simplify this expression:
[tex]\[ 3 \cdot 3^{x-1} = 3^1 \cdot 3^{x-1} = 3^{1 + (x-1)} = 3^x \][/tex]
So, [tex]\( 3 \cdot 3^{x-1} \)[/tex] is equivalent to [tex]\( 3^x \)[/tex].
### Option B: [tex]\( \left( \frac{18}{6} \right)^x \)[/tex]
Simplify the base inside the parentheses:
[tex]\[ \frac{18}{6} = 3 \][/tex]
Thus,
[tex]\[ \left( \frac{18}{6} \right)^x = 3^x \][/tex]
So, [tex]\( \left( \frac{18}{6} \right)^x \)[/tex] is equivalent to [tex]\( 3^x \)[/tex].
### Option C: [tex]\( \frac{18^x}{6^x} \)[/tex]
We can use the rules of exponents to simplify this expression:
[tex]\[ \frac{18^x}{6^x} = \left(\frac{18}{6}\right)^x = 3^x \][/tex]
So, [tex]\( \frac{18^x}{6^x} \)[/tex] is equivalent to [tex]\( 3^x \)[/tex].
### Option D: [tex]\( 3 \cdot 3^{x+1} \)[/tex]
We can simplify this expression using the rules of exponents:
[tex]\[ 3 \cdot 3^{x+1} = 3^1 \cdot 3^{x+1} = 3^{1 + (x+1)} = 3^{x+2} \][/tex]
This is not equivalent to [tex]\( 3^x \)[/tex].
### Option E: [tex]\( x^3 \)[/tex]
This expression cannot be simplified to match [tex]\( 3^x \)[/tex]. They are fundamentally different forms. So, [tex]\( x^3 \)[/tex] is not equivalent to [tex]\( 3^x \)[/tex].
### Option F: [tex]\( \frac{18^x}{3} \)[/tex]
We can rewrite the denominator with an exponent:
[tex]\[ \frac{18^x}{3} = \frac{18^x}{3^1} \][/tex]
This can also be written as:
[tex]\[ \frac{18^x}{3^1} = \frac{18^x}{3} \][/tex]
There is no way to further simplify this to match [tex]\( 3^x \)[/tex]. So, [tex]\( \frac{18^x}{3} \)[/tex] is not equivalent to [tex]\( 3^x \)[/tex].
### Summary
From the above analysis, we find that the following expressions are equivalent to [tex]\( 3^x \)[/tex]:
- Option A: [tex]\( 3 \cdot 3^{x-1} \)[/tex]
- Option C: [tex]\( \frac{18^x}{6^x} \)[/tex]
Therefore, the expressions equivalent to [tex]\( 3^x \)[/tex] are A and C.
### Option A: [tex]\( 3 \cdot 3^{x-1} \)[/tex]
We can use the rules of exponents to simplify this expression:
[tex]\[ 3 \cdot 3^{x-1} = 3^1 \cdot 3^{x-1} = 3^{1 + (x-1)} = 3^x \][/tex]
So, [tex]\( 3 \cdot 3^{x-1} \)[/tex] is equivalent to [tex]\( 3^x \)[/tex].
### Option B: [tex]\( \left( \frac{18}{6} \right)^x \)[/tex]
Simplify the base inside the parentheses:
[tex]\[ \frac{18}{6} = 3 \][/tex]
Thus,
[tex]\[ \left( \frac{18}{6} \right)^x = 3^x \][/tex]
So, [tex]\( \left( \frac{18}{6} \right)^x \)[/tex] is equivalent to [tex]\( 3^x \)[/tex].
### Option C: [tex]\( \frac{18^x}{6^x} \)[/tex]
We can use the rules of exponents to simplify this expression:
[tex]\[ \frac{18^x}{6^x} = \left(\frac{18}{6}\right)^x = 3^x \][/tex]
So, [tex]\( \frac{18^x}{6^x} \)[/tex] is equivalent to [tex]\( 3^x \)[/tex].
### Option D: [tex]\( 3 \cdot 3^{x+1} \)[/tex]
We can simplify this expression using the rules of exponents:
[tex]\[ 3 \cdot 3^{x+1} = 3^1 \cdot 3^{x+1} = 3^{1 + (x+1)} = 3^{x+2} \][/tex]
This is not equivalent to [tex]\( 3^x \)[/tex].
### Option E: [tex]\( x^3 \)[/tex]
This expression cannot be simplified to match [tex]\( 3^x \)[/tex]. They are fundamentally different forms. So, [tex]\( x^3 \)[/tex] is not equivalent to [tex]\( 3^x \)[/tex].
### Option F: [tex]\( \frac{18^x}{3} \)[/tex]
We can rewrite the denominator with an exponent:
[tex]\[ \frac{18^x}{3} = \frac{18^x}{3^1} \][/tex]
This can also be written as:
[tex]\[ \frac{18^x}{3^1} = \frac{18^x}{3} \][/tex]
There is no way to further simplify this to match [tex]\( 3^x \)[/tex]. So, [tex]\( \frac{18^x}{3} \)[/tex] is not equivalent to [tex]\( 3^x \)[/tex].
### Summary
From the above analysis, we find that the following expressions are equivalent to [tex]\( 3^x \)[/tex]:
- Option A: [tex]\( 3 \cdot 3^{x-1} \)[/tex]
- Option C: [tex]\( \frac{18^x}{6^x} \)[/tex]
Therefore, the expressions equivalent to [tex]\( 3^x \)[/tex] are A and C.
