Which is equivalent to [tex]$\sqrt[5]{1215}^x$[/tex]?

A. [tex]$243^x$[/tex]

B. [tex]$1215^{\frac{1}{5} x}$[/tex]

C. [tex][tex]$1215^{\frac{1}{5x}}$[/tex][/tex]

D. [tex]$243^{\frac{1}{x}}$[/tex]



Answer :

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]