To determine which expression is equivalent to [tex]\(\sqrt[5]{1,215}^x\)[/tex], we need to understand the relationship between radical and exponential forms.
1. Let's start by rewriting [tex]\(\sqrt[5]{1,215}\)[/tex] in an exponential form. The fifth root of [tex]\(1,215\)[/tex] can be expressed as:
[tex]\[
\sqrt[5]{1,215} = 1,215^{\frac{1}{5}}
\][/tex]
2. Now, if we raise this to the power of [tex]\(x\)[/tex], we get:
[tex]\[
\left(1,215^{\frac{1}{5}}\right)^x
\][/tex]
3. According to the rules of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Therefore, we can simplify the expression:
[tex]\[
\left(1,215^{\frac{1}{5}}\right)^x = 1,215^{\frac{1}{5} \cdot x}
\][/tex]
4. By examining the choices given:
- [tex]\(243^x\)[/tex]
- [tex]\(1,215^{\frac{1}{5} x}\)[/tex]
- [tex]\(1,215^{\frac{1}{5 x}}\)[/tex]
- [tex]\(243^{\frac{1}{x}}\)[/tex]
It is evident that the only choice matching our simplified expression [tex]\(1,215^{\frac{1}{5} x}\)[/tex] is
[tex]\(\boxed{1,215^{\frac{1}{5} x}}\)[/tex]
