When the stated relation below is true?

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OmarMohamedKhallaf

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The question goes as follows:
The relation [MATH]\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}[/MATH] is true for all values ....
  1. [MATH]a\in\mathbb{R}[/MATH]
  2. [MATH]n\in\mathbb{N}^{+}[/MATH]
  3. [MATH]\sqrt[n]{a}\in\mathbb{R}, n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH]
  4. Nothing from the previous
The book says that the answer is no. 3, but I don't understand why and I think this is not right.
I think that this part [MATH]n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH] is not necessary.
 
OmarMohamedKhallaf said:
The question goes as follows:
The relation [MATH]\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}[/MATH] is true for all values ....
  1. [MATH]a\in\mathbb{R}[/MATH]
  2. [MATH]n\in\mathbb{N}^{+}[/MATH]
  3. [MATH]\sqrt[n]{a}\in\mathbb{R}, n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH]
  4. Nothing from the previous
The book says that the answer is no. 3, but I don't understand why and I think this is not right.
I think that this part [MATH]n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH] is not necessary.
Click to expand...
Pick an example with a negative n and check.
 
I'm not sure I agree with them about excluding n=1; isn't the first root of a just a? Or does this book just not allow a first root for some reason?
 
OmarMohamedKhallaf said:
The book says that the answer is no. 3, but I don't understand why and I think this is not right.
I think that this part [MATH]n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH] is not necessary.
Click to expand...

If dogs can bark, then saying that poodles can bark is also a true statement.
(However saying that only poodles can bark would be wrong.)

If a function accepts 0<x<10, then it is also true to say it accepts 2<x<5.

OP is right that the book could have missed off the extra constraint, but their option 3 is nonetheless true (even if this doesn't convey that the relation also works for other numbers.)
 
Cubist said:
If dogs can bark, then saying that poodles can bark is also a true statement.
(However saying that only poodles can bark would be wrong.)

If a function accepts 0<x<10, then it is also true to say it accepts 2<x<5.

OP is right that the book could have missed off the extra constraint, but their option 3 is nonetheless true (even if this doesn't convey that the relation also works for other numbers.)
Click to expand...
Good point. I had observed that the question doesn't ask for a necessary and sufficient condition, only for a sufficient condition; but I hadn't applied that observation to (3).

So we don't have to explain why 1 is excluded, and it doesn't matter that any one part is not necessary.
OmarMohamedKhallaf said:
The question goes as follows:
The relation [MATH]\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}[/MATH] is true for all values ....
  1. [MATH]a\in\mathbb{R}[/MATH]
  2. [MATH]n\in\mathbb{N}^{+}[/MATH]
  3. [MATH]\sqrt[n]{a}\in\mathbb{R}, n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH]
  4. Nothing from the previous
The book says that the answer is no. 3, but I don't understand why and I think this is not right.
I think that this part [MATH]n\in\mathbb{N}^{+}-\left\{1\right\}[/MATH] is not necessary.
Click to expand...

The answer is not (1), because there are real numbers a for which it is not always true; it is not (2) because there are positive integers n for which it is not always true; but it is (3), because under those conditions it is always true.
 
Cubist said:
If dogs can bark, then saying that poodles can bark is also a true statement.
(However saying that only poodles can bark would be wrong.)

If a function accepts 0<x<10, then it is also true to say it accepts 2<x<5.

OP is right that the book could have missed off the extra constraint, but their option 3 is nonetheless true (even if this doesn't convey that the relation also works for other numbers.)
Click to expand...
You are correct and, IMHO, that's one nasty thing to do to a student!

-Dan
 
lev888 said:
Pick an example with a negative n and check.
Click to expand...
I did that, and here are the test cases:
[MATH] \text{let }a=2, n = \frac{-1}{2}, m = 3 \\ \therefore \sqrt[\frac{-1}{2}]{2^{3}} = 0.015625 \\ \because \text{By definition } \sqrt[n]{a} = a^{\frac{1}{n}} \\ \therefore \sqrt[\frac{-1}{2}]{2^{3}}=2^{-\frac{1}{\frac{1}{2}}\times3}=2^{-2\times3} = 0.015626 [/MATH]
 
Cubist said:
If dogs can bark, then saying that poodles can bark is also a true statement.
(However saying that only poodles can bark would be wrong.)

If a function accepts 0<x<10, then it is also true to say it accepts 2<x<5.

OP is right that the book could have missed off the extra constraint, but their option 3 is nonetheless true (even if this doesn't convey that the relation also works for other numbers.)
Click to expand...
Yes, I think this is the right way to interpret why this might be the right answer.
 
Dr.Peterson said:
Good point. I had observed that the question doesn't ask for a necessary and sufficient condition, only for a sufficient condition; but I hadn't applied that observation to (3).

So we don't have to explain why 1 is excluded, and it doesn't matter that any one part is not necessary.


The answer is not (1), because there are real numbers a for which it is not always true; it is not (2) because there are positive integers n for which it is not always true; but it is (3), because under those conditions it is always true.
Click to expand...
Yes, Excluding other options is will leave me with option 3 only, but I was confused by the extra constraints. I thought I was missing some ideas.
 
firemath said:
The title of this thread is silly.
Click to expand...
firemath, I am getting a bit concerned that sometimes you say negative things towards the students who come here and I don't think that there is a reason most times for you. Complaining about a title, what value does anyone get from you saying that? Some students may not return because of that.
I am not saying that I am always nice to students but your complaints seem to have no value. Please understand that the students who come here are our students and we need to help them and hope that they come back.
 
Jomo said:
firemath, I am getting a bit concerned that sometimes you say negative things towards the students who come here and I don't think that there is a reason most times for you. Complaining about a title, what value does anyone get from you saying that? Some students may not return because of that.
I am not saying that I am always nice to students but your complaints seem to have no value. Please understand that the students who come here are our students and we need to help them and hope that they come back.
Click to expand...
Please look at my reply to OP.
 
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