Sure, let's analyze each statement one by one, given that [tex]\( a^{1/n} = r \)[/tex].
### Statement A: [tex]\( r^n = a \)[/tex]
To verify this, we start with the given equation:
[tex]\[ a^{1/n} = r \][/tex]
Raise both sides to the power of [tex]\( n \)[/tex]:
[tex]\[ (a^{1/n})^n = r^n \][/tex]
The left side simplifies as follows:
[tex]\[ a^{(1/n) \cdot n} = r^n \][/tex]
[tex]\[ a^1 = r^n \][/tex]
[tex]\[ a = r^n \][/tex]
So, this statement is indeed true.
### Statement B: [tex]\( a^r = n \)[/tex]
Using the given equation [tex]\( a^{1/n} = r \)[/tex], let's see if this implies [tex]\( a^r = n \)[/tex].
Rewriting [tex]\( a \)[/tex] in terms of [tex]\( r \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ r = a^{1/n} \][/tex]
Equivalently:
[tex]\[ r \neq a^r = n \][/tex]
There is no direct algebraic relationship that shows [tex]\( a^r = n \)[/tex] given [tex]\( a^{1/n} = r \)[/tex]. Therefore, this statement is false.
### Statement C: [tex]\( \sqrt{\sqrt{a}} = r \)[/tex]
The statement [tex]\( \sqrt{\sqrt{a}} \)[/tex] can be expressed as:
[tex]\[ \sqrt{\sqrt{a}} = (a^{1/2})^{1/2} \][/tex]
[tex]\[ = a^{(1/2) \cdot (1/2)} = a^{1/4} \][/tex]
The given equation is [tex]\( a^{1/n} = r \)[/tex]. For the statement to be true:
[tex]\[ a^{1/n} = a^{1/4} \][/tex]
This implies:
[tex]\[ \frac{1}{n} = \frac{1}{4} \][/tex]
[tex]\[ n = 4 \][/tex]
However, this is only true if [tex]\( n = 4 \)[/tex], which is not necessarily the case for all [tex]\( n \)[/tex]. Thus, this statement is false.
### Statement D: [tex]\( n^{1/r} = a \)[/tex]
Using the given equation:
[tex]\[ a^{1/n} = r \][/tex]
Taking the [tex]\( n \)[/tex]-th root on both sides:
[tex]\[ r^n = a \][/tex]
Rewriting [tex]\( r \)[/tex] in terms of [tex]\( a \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ r = a^{1/n} \][/tex]
Now consider the statement [tex]\( n^{1/r} = a \)[/tex]. Rewriting, we have:
[tex]\[ n^{1/(a^{1/n})} \neq a \][/tex]
This does not directly follow from the equation [tex]\( a^{1/n} = r \)[/tex]. Therefore, this statement is false.
### Summary:
- Statement A: [tex]\( r^n = a \)[/tex] — True
- Statement B: [tex]\( a^r = n \)[/tex] — False
- Statement C: [tex]\( \sqrt{\sqrt{a}} = r \)[/tex] — False
- Statement D: [tex]\( n^{1/r} = a \)[/tex] — False
Therefore, the true statements are:
[tex]\[ A. \quad r^n = a \][/tex]
The other statements are false.
