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May 6th, 2014, 04:07 AM   #1
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Lightbulb Why not these surds are equal?

Okay, there is a question in my text book.

Which one is true?
A)$\displaystyle \sqrt{2}=\sqrt{(-1)(-2)}=\sqrt{(-1)}.\sqrt{(-2)}$
B)$\displaystyle \sqrt{4}=\sqrt{(-2)(-2)}=\sqrt{2}.\sqrt{2}=2$
C)$\displaystyle \sqrt{4}=\sqrt{(-2)(-2)}=\left (- \sqrt{2} \right )\left (- \sqrt{2} \right )=2$
D)$\displaystyle \sqrt{4}=\sqrt{(-2)(-2)}=\sqrt{2*2}=2$

Okay, Option a is false because square root of negative number results in imaginary number and goes on.

Option b. Both I and text book accept it true.

Option c. I accept even it is also true along with b.
$\displaystyle \\
\sqrt{4}=\sqrt{(-2)(-2)}=\left (- \sqrt{2} \right )\left (- \sqrt{2} \right )=2\\
=\sqrt{4}=\sqrt{(+4)}=+\left (\sqrt{2} \right )^2=2\\
=\sqrt{4}=\sqrt{4}=2=2$

I think even option d is also right.
$\displaystyle \\
\sqrt{4}=\sqrt{(-2)(-2)}=\sqrt{2*2}=2\\
=\sqrt{4}=\sqrt{+(4)}=\sqrt{4}=2\\
=\sqrt{4}=\sqrt{4}=\sqrt{4}=2\\$

But my text book says only option b is right. There's no explanation. Is only option b right?Or is option b, c, d right? Or is all of them right?

Last edited by GovindBalaji; May 6th, 2014 at 04:28 AM.
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May 6th, 2014, 05:53 AM   #2
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In my opinion, only option A is false. Options B, C, D are all true.
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May 6th, 2014, 06:05 AM   #3
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Thumbs up ????

Quote:
Originally Posted by Olinguito View Post
In my opinion, only option A is false. Options B, C, D are all true.
That's what I also think.
But why only option b is there in book.
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May 6th, 2014, 06:28 AM   #4
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Choice c seems to suggest $\displaystyle \sqrt{-2}=-\sqrt2$ which isn't true.
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May 6th, 2014, 06:57 AM   #5
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Nevertheless $\sqrt{(-a)(-a)}=(-\sqrt a)(-\sqrt a)$ ($a\in\mathbb R^+\cup\{0\}$) is true.

Whether a statement is true or not has nothing to do with what it “seems to suggest”.
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Last edited by Olinguito; May 6th, 2014 at 07:17 AM.
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May 6th, 2014, 07:18 AM   #6
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Sure, but it's not clear what is intended (at least to me). Even though what you say is true, $\displaystyle \sqrt{-2}=-\sqrt{2}$ is not true. Most equalities are constructed to show a mathematical progression from one idea to another, are they not?

Last edited by greg1313; May 6th, 2014 at 07:37 AM.
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May 6th, 2014, 07:56 AM   #7
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It's a rubbish question in my opinion. Poke your teacher and get him/her to set you better questions or get a different textbook!
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