Answer :
To address the question of whether the statement [tex]\( c^{1/n} = \sqrt{c} \)[/tex] is true for any nonnegative real number [tex]\( c \)[/tex] and any positive integer [tex]\( n \)[/tex], let’s discuss it in detail.
### Understanding the Exponentiation and Roots
1. Notation and Definitions:
- [tex]\( c^{1/n} \)[/tex]: This represents the [tex]\( n \)[/tex]-th root of [tex]\( c \)[/tex].
- [tex]\( \sqrt{c} \)[/tex]: This typically represents the square root of [tex]\( c \)[/tex], which can also be written as [tex]\( c^{1/2} \)[/tex].
2. Special Case for [tex]\( n = 2 \)[/tex]:
- If [tex]\( n = 2 \)[/tex], the equation [tex]\( c^{1/n} = c^{1/2} \)[/tex] simplifies to [tex]\( \sqrt{c} = \sqrt{c} \)[/tex], which is true by definition.
3. General Case for [tex]\( n \neq 2 \)[/tex]:
- For [tex]\( n \neq 2 \)[/tex], [tex]\( c^{1/n} \)[/tex] is not necessarily equal to [tex]\( \sqrt{c} \)[/tex]:
- For example, if [tex]\( c = 16 \)[/tex] and [tex]\( n = 4 \)[/tex]:
- [tex]\( c^{1/4} \)[/tex] is the fourth root of [tex]\( 16 \)[/tex], which is [tex]\( 2 \)[/tex].
- [tex]\( \sqrt{16} \)[/tex] is the square root of [tex]\( 16 \)[/tex], which is [tex]\( 4 \)[/tex].
- Clearly, [tex]\( 2 \neq 4 \)[/tex], so this shows that [tex]\( c^{1/n} \neq \sqrt{c} \)[/tex] when [tex]\( n = 4 \)[/tex].
- Hence, the statement [tex]\( c^{1/n} = \sqrt{c} \)[/tex] does not hold for [tex]\( n = 4 \)[/tex], and similar discrepancies will occur for other values of [tex]\( n \neq 2 \)[/tex].
### Conclusion
After carefully examining specific values and considering the definitions, we conclude that the statement:
[tex]\[ c^{1/n} = \sqrt{c} \][/tex]
is false in general. It only holds true specifically when [tex]\( n = 2 \)[/tex]. Because the question does not limit [tex]\( n \)[/tex] to just 2, the overall statement is:
B. False
To summarize, the result from our verification shows that the statement is false in the general case. For [tex]\( c = 16 \)[/tex] and [tex]\( n = 4 \)[/tex]:
- [tex]\( c^{1/4} = 2.0 \)[/tex]
- [tex]\( \sqrt{c} = 4.0 \)[/tex]
These are not equal, confirming the statement is false. However, when [tex]\( n = 2 \)[/tex]:
- [tex]\( c^{1/2} = 4.0 \)[/tex]
- [tex]\( \sqrt{16} = 4.0 \)[/tex]
These are equal, but since the statement must be true for all [tex]\( n \)[/tex] to be true, and we found a case where it is not, the final answer is:
B. False
### Understanding the Exponentiation and Roots
1. Notation and Definitions:
- [tex]\( c^{1/n} \)[/tex]: This represents the [tex]\( n \)[/tex]-th root of [tex]\( c \)[/tex].
- [tex]\( \sqrt{c} \)[/tex]: This typically represents the square root of [tex]\( c \)[/tex], which can also be written as [tex]\( c^{1/2} \)[/tex].
2. Special Case for [tex]\( n = 2 \)[/tex]:
- If [tex]\( n = 2 \)[/tex], the equation [tex]\( c^{1/n} = c^{1/2} \)[/tex] simplifies to [tex]\( \sqrt{c} = \sqrt{c} \)[/tex], which is true by definition.
3. General Case for [tex]\( n \neq 2 \)[/tex]:
- For [tex]\( n \neq 2 \)[/tex], [tex]\( c^{1/n} \)[/tex] is not necessarily equal to [tex]\( \sqrt{c} \)[/tex]:
- For example, if [tex]\( c = 16 \)[/tex] and [tex]\( n = 4 \)[/tex]:
- [tex]\( c^{1/4} \)[/tex] is the fourth root of [tex]\( 16 \)[/tex], which is [tex]\( 2 \)[/tex].
- [tex]\( \sqrt{16} \)[/tex] is the square root of [tex]\( 16 \)[/tex], which is [tex]\( 4 \)[/tex].
- Clearly, [tex]\( 2 \neq 4 \)[/tex], so this shows that [tex]\( c^{1/n} \neq \sqrt{c} \)[/tex] when [tex]\( n = 4 \)[/tex].
- Hence, the statement [tex]\( c^{1/n} = \sqrt{c} \)[/tex] does not hold for [tex]\( n = 4 \)[/tex], and similar discrepancies will occur for other values of [tex]\( n \neq 2 \)[/tex].
### Conclusion
After carefully examining specific values and considering the definitions, we conclude that the statement:
[tex]\[ c^{1/n} = \sqrt{c} \][/tex]
is false in general. It only holds true specifically when [tex]\( n = 2 \)[/tex]. Because the question does not limit [tex]\( n \)[/tex] to just 2, the overall statement is:
B. False
To summarize, the result from our verification shows that the statement is false in the general case. For [tex]\( c = 16 \)[/tex] and [tex]\( n = 4 \)[/tex]:
- [tex]\( c^{1/4} = 2.0 \)[/tex]
- [tex]\( \sqrt{c} = 4.0 \)[/tex]
These are not equal, confirming the statement is false. However, when [tex]\( n = 2 \)[/tex]:
- [tex]\( c^{1/2} = 4.0 \)[/tex]
- [tex]\( \sqrt{16} = 4.0 \)[/tex]
These are equal, but since the statement must be true for all [tex]\( n \)[/tex] to be true, and we found a case where it is not, the final answer is:
B. False
