To determine whether the statement [tex]\( b^{1 / n} = \sqrt[n]{b} \)[/tex] is true or false, let's carefully explore the concepts involved.
1. Understanding [tex]\( b^{1 / n} \)[/tex]:
- [tex]\( b \)[/tex] is a nonnegative real number, meaning [tex]\( b \geq 0 \)[/tex].
- [tex]\( n \)[/tex] is a positive integer, meaning [tex]\( n \)[/tex] is a whole number greater than zero.
- [tex]\( b^{1 / n} \)[/tex] means taking the [tex]\( n \)[/tex]-th root of [tex]\( b \)[/tex], which is equivalent to raising [tex]\( b \)[/tex] to the power of [tex]\( 1 / n \)[/tex].
2. Understanding [tex]\( \sqrt[n]{b} \)[/tex]:
- The expression [tex]\( \sqrt[n]{b} \)[/tex] denotes the [tex]\( n \)[/tex]-th root of [tex]\( b \)[/tex].
- This is the number that, when raised to the power [tex]\( n \)[/tex], yields [tex]\( b \)[/tex].
3. Equivalence of the Expressions:
- The notation [tex]\( b^{1 / n} \)[/tex] is another way of writing [tex]\( \sqrt[n]{b} \)[/tex].
- Both expressions perform the same mathematical operation: taking the [tex]\( n \)[/tex]-th root of [tex]\( b \)[/tex].
4. Example Verification:
- If [tex]\( b = 4 \)[/tex] and [tex]\( n = 2 \)[/tex], then:
- [tex]\( b^{1 / n} = 4^{1 / 2} = \sqrt{4} = 2 \)[/tex].
- [tex]\( \sqrt[n]{b} = \sqrt[2]{4} = \sqrt{4} = 2 \)[/tex].
With the above step-by-step explanation, it becomes clear that the expression [tex]\( b^{1 / n} \)[/tex] is indeed equivalent to [tex]\( \sqrt[n]{b} \)[/tex].
Therefore, the statement [tex]\( b^{1 / n} = \sqrt[n]{b} \)[/tex] is:
A. True
