To determine which statements are true given the equation [tex]\(a^{1 / n} = r\)[/tex], we need to analyze each provided statement. Let's examine them one by one:
### Statement A: [tex]\(^n \sqrt{ } a=r\)[/tex]
The notation [tex]\(^n \sqrt{ } a\)[/tex] represents the [tex]\(n\)[/tex]-th root of [tex]\(a\)[/tex]. This is mathematically equivalent to [tex]\(a^{1/n}\)[/tex]. Since the given condition is [tex]\(a^{1/n} = r\)[/tex], this statement is true. Therefore, Statement A is true.
### Statement B: [tex]\(r^n = a\)[/tex]
To verify if this statement is true, we can raise both sides of the equation [tex]\(a^{1/n} = r\)[/tex] to the power of [tex]\(n\)[/tex]:
[tex]\[
(a^{1/n})^n = r^n
\][/tex]
Simplifying the left side:
[tex]\[
a^{(1/n) \cdot n} = r^n \implies a = r^n
\][/tex]
Thus, [tex]\(r^n = a\)[/tex] is a direct consequence of the initial condition. Therefore, Statement B is true.
### Statement C: [tex]\(n^{1/T!}=a\)[/tex]
This statement seems to introduce unrelated variables and factors into the equation. Specifically, it uses [tex]\(T!\)[/tex], the factorial of [tex]\(T\)[/tex], and relates it to [tex]\(n\)[/tex] and [tex]\(a\)[/tex]. There is no basis in the given equation [tex]\(a^{1/n} = r\)[/tex] to support this relationship. Therefore, Statement C is not related and hence is false.
### Statement D: [tex]\(d = n\)[/tex]
The statement [tex]\(d = n\)[/tex] introduces a new variable [tex]\(d\)[/tex] that is not mentioned or related in the given equation. There is no information provided about [tex]\(d\)[/tex] or its relationship to [tex]\(n\)[/tex]. Thus, this statement cannot be verified from the given information and is not relevant. Therefore, Statement D is false.
### Summary
From the analysis, the true statements based on the given equation [tex]\(a^{1/n} = r\)[/tex] are:
- A: True
- B: True
Therefore, Statements A and B are correct.
