Hi,

So I'm hoping someone can clarify for me why it is possible to take out the negative sign that is under the radical (in the image attached)? I thought this was only possible if the exponent which the number was raised to was odd. e.g. I thought that (-27)^(1/3) = -(27)^(1/3) is allowed, but (-27)^(1/2) = -(27)^(1/2), would not be.

- otownsend

2. Re: negative sign under radical

Do you know what imaginary numbers are?

3. Re: negative sign under radical

yes, but the course hasn't covered imaginary numbers yet, so I'm assuming that this is a mistake. Will this question work only in the context of imaginary numbers?

4. Re: negative sign under radical

There are two errors in the attachment:

1. sqrt(-27) does not equal -1 times sqrt(27).
2. sqrt(27) is not 3.

You can tell that it's wrong by testing the final result: if you try a number such as x = -4 in the original equation you get -3x^2 = -3 x (-4)^2 = 3 x 16 = 48, and that is not >= 81.

The correct steps are:

-3x^2 >= 81
x^2 <= -27
x <= sqrt(-27)
x <= sqrt(27) i, where i = sqrt(-1).

So the result requires an understanding of imaginary numbers.

ok thanks

6. Re: negative sign under radical

Solve pretending there is no minus sign, then insert the "i"...

7. Re: negative sign under radical

Originally Posted by otownsend
yes, but the course hasn't covered imaginary numbers yet, so I'm assuming that this is a mistake. Will this question work only in the context of imaginary numbers?
From my experience, books are riddled with mistakes let alone some course. It's down to YOU to notice the mistakes.

Assumption is the mother of all ups.

8. Re: negative sign under radical

Originally Posted by PilgrimsPath
From my experience, books are riddled with mistakes let alone some course. It's down to YOU to notice the mistakes.

Assumption is the mother of all ups.
She did notice the mistake. And then she came here asking for clarification.

Sounds like a pretty effective set of behaviors to me.

9. Re: negative sign under radical

Originally Posted by otownsend
clarify for me why it is possible to take out the negative sign that is under the radical (in the image attached)? I thought this was only possible if the exponent which the number was raised to was odd. e.g. I thought that (-27)^(1/3) = -(27)^(1/3) is allowed, but (-27)^(1/2) = -(27)^(1/2), would not be.
If you assume that all variables are real (there are no complex number involved) then $\bf{(\forall x)[-3x^2\le 0]}$.
So simply by inspection $\bf{-3x^2\ge 81}$ cannot possibly have a solution.
In the testing community that type question is known as a turtle.
It is designed to slow down a poorly prepared test taker.
In other words, once the question is read there is absolutely no reason not to know the answer at once.

10. Re: negative sign under radical

Originally Posted by romsek
She did notice the mistake. And then she came here asking for clarification.

Sounds like a pretty effective set of behaviors to me.
Yes, my argument coincided with her reasoning. It was in agreement (not denial) with what she claimed.

Amateur.

11. Re: negative sign under radical

Thinking about this, it occurs to me that the root of the problem (excuse the pun) is that the first line of the problem statement has a typo. Instead of -3x^2 >= 81, if that first line was -3x^3 >= 81 then rest works out just as presented:

-3x^3 >= 81
x^3 <= -27
cube_root {x^3} <= cube_root {-27}
cube_root {x^3} <= cube_root {(-1)^3 (27)}

Now you can pull the -1 out front of the cube root:

x^3 <= - cube_root {27}
x <= -3

To the OP: please check whether this was indeed the problem statement.