Are these two expressions equivalent or not?

[poolzey89]

and

Can somebody help me out in trying to explain why they are not the same (I think).

If we square both expressions, we get for both, but then if we take the square root, we will be left with only the expression on the right.

I am a bit confused about taking the positive and negative square roots, so if someone could provide an easy to understand explanation about what is correct, that would really help me out, because it's just bugging me at the moment.

 

[lev888]

View attachment 17639 and View attachment 17640

Can somebody help me out in trying to explain why they are not the same (I think).

If we square both expressions, we get View attachment 17641 for both, but then if we take the square root, we will be left with only the expression on the right.

I am a bit confused about taking the positive and negative square roots, so if someone could provide an easy to understand explanation about what is correct, that would really help me out, because it's just bugging me at the moment.

Both expressions are positive, why do you need to consider negative roots?
When taking the square root you get the first expression if you follow specific steps, namely, what you did during its squaring, in reverse.

 

[Dr.Peterson]

Both are equal to 1.0352761804100830493955953504962..., so yes, they are equivalent.

Why do you think two expressions written differently can't have the same value?

 

[poolzey89]

Both are equal to 1.0352761804100830493955953504962..., so yes, they are equivalent.

Why do you think two expressions written differently can't have the same value?

I think I was just getting confused with going in the opposite direction. E.g

Thanks for both your help.

 

[Dr.Peterson]

Ok, let's look at an example where something actually does go wrong if you don't think carefully. Suppose I want to rewrite [MATH]\sqrt{2} - \sqrt{6}[/MATH], so I square it. I get [MATH]2 - 2\sqrt{12} + 6 = 8 - 4\sqrt{3}[/MATH], so I figure that [MATH]\sqrt{2} - \sqrt{6} = \sqrt{8 - 4\sqrt{3}}[/MATH].

The only thing I did that could be wrong is when I assumed that the only number whose square is [MATH]8 - 4\sqrt{3}[/MATH] is its square root. I actually have to consider two possibilities, that it is [MATH]\sqrt{8 - 4\sqrt{3}}[/MATH] or [MATH]-\sqrt{8 - 4\sqrt{3}}[/MATH]. (Any number whose square is [MATH]8 - 4\sqrt{3}[/MATH] must be one or the other.) To tell the difference, I check the sign of [MATH]\sqrt{2} - \sqrt{6}[/MATH], and realize that it is negative, so the second answer is the correct one.

 

[poolzey89]

Ok, let's look at an example where something actually does go wrong if you don't think carefully. Suppose I want to rewrite [MATH]\sqrt{2} - \sqrt{6}[/MATH], so I square it. I get [MATH]2 - 2\sqrt{12} + 6 = 8 - 4\sqrt{3}[/MATH], so I figure that [MATH]\sqrt{2} - \sqrt{6} = \sqrt{8 - 4\sqrt{3}}[/MATH].

The only thing I did that could be wrong is when I assumed that the only number whose square is [MATH]8 - 4\sqrt{3}[/MATH] is its square root. I actually have to consider two possibilities, that it is [MATH]\sqrt{8 - 4\sqrt{3}}[/MATH] or [MATH]-\sqrt{8 - 4\sqrt{3}}[/MATH]. (Any number whose square is [MATH]8 - 4\sqrt{3}[/MATH] must be one or the other.) To tell the difference, I check the sign of [MATH]\sqrt{2} - \sqrt{6}[/MATH], and realize that it is negative, so the second answer is the correct one.

Thank you very much for this detailed and thorough explanation - that makes perfect sense.