To determine which of the choices is equivalent to the equation [tex]\(3^{(x-1)} = 27\)[/tex], let's examine the choices one by one.
First, we need to express 27 as a power of 3. Since:
[tex]\[ 27 = 3^3 \][/tex]
We can substitute this into the original equation:
[tex]\[ 3^{(x-1)} = 27 \][/tex]
[tex]\[ 3^{(x-1)} = 3^3 \][/tex]
We now have:
[tex]\[ 3^{(x-1)} = 3^3 \][/tex]
This shows that the equation [tex]\(3^{(x-1)} = 27\)[/tex] is equivalent to [tex]\(3^{(x-1)} = 3^3\)[/tex].
Let's check the other choices to see if any of them are correct:
- [tex]\(3^{(x-1)} = 3^{-3}\)[/tex]: This cannot be equivalent because 27 is a positive number and [tex]\(3^{-3} = \frac{1}{27}\)[/tex], which is not equal to 27.
- [tex]\(3^{(x-1)} = 9^3\)[/tex]: Here, [tex]\(9 = 3^2\)[/tex] so [tex]\(9^3 = (3^2)^3 = 3^6\)[/tex]. This is clearly not the same as [tex]\(3^3\)[/tex].
Thus, the correct choice equivalent to the equation [tex]\(3^{(x-1)} = 27\)[/tex] is:
[tex]\[ \boxed{3^{(x-1)} = 3^3} \][/tex]
