Roots of Multi-Term Expressions: Simplifying and Representing √(a^2 + b^2)

[Atran]

Hi, How can I write, for instance, the square root of (a² + b²) in another form?
Is there another expression which equals √(a² + b²), and is it possible for that expression to have its √() holding one term whereas other factors exists outside the √() in that expression...

Thanks...

 

[Mark44]

Hi, How can I write, for instance, the square root of (a² + b²) in another form?

$$\sqrt{a^2 + b^2}$$
can't be further simplified without knowing the values of a and b.

Is there another expression which equals √(a² + b²), and is it possible for that expression to have its √() holding one term whereas other factors exists outside the √() in that expression...

Thanks...

 

[Gerenuk]

You can make up something like
$$\sqrt{a^2+b^2}=\sqrt{a^2+\sqrt{a^4-b^4/4}}+\sqrt{a^2-\sqrt{a^4-b^4/4}}$$
but there is usually no useful simplification.

Obviously you cannot have
$$\sqrt{a^2+b^2}=f(a)+g(b)$$
since if you differentiate both sides by a and then by b, then you get zero on the right, but something else on the left.