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May 21st, 2009, 03:56 PM   #1
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simplifying sqrt[a + sqrt(b)]

[color=#000080]Greetings:

Is there a means by which to simplify the likes of sqrt[2 + sqrt(3)]. That is, can we rewrite this with a single radical?

Thanks,

Rich B.[/color]
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May 21st, 2009, 06:50 PM   #2
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Re: simplifying sqrt[a + sqrt(b)]

Generally speaking, no. Sorry!
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May 22nd, 2009, 03:11 AM   #3
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Re: simplifying sqrt[a + sqrt(b)]

Quote:
Originally Posted by nikkor180
[color=#000080]Greetings:

Is there a means by which to simplify the likes of sqrt[2 + sqrt(3)]. That is, can we rewrite this with a single radical?

Thanks,

Rich B.[/color]
I am not aware of a general methods, but a little guesswork can help quite often.

For instance you can try to look for (or any form which seems plausible) fulfilling and compare coefficients at the roots.

The expression can be simplified as follows: Notice (this is the initial guess) that

This yields


Sometimes you can also use some trigonometric identities. We have


Now as see here we get

which is the same result as before. (Although, I do not see any quick derivation of the formula for that I copied from wikipedia.)
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June 25th, 2009, 04:18 AM   #4
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Re: simplifying sqrt[a + sqrt(b)]

You can also check whether its minimal polynomial is reducable.
is a root of the polynomial

If this polynomial is irreducable then is its minimum polynomial and this will not help you to
to simplify the expression .

However if can be reduced to a product of 2 polynomials of degree 2, say , then is either
a root of or . It's then easy to check with the abc-formula wich of the 4 roots correspond with .

To give an example: is a root of the polynomal
Notice that can be reduced to , so is either a root of g or h. By checking all roots of g and h with the abc-formula: (roots of g) and (roots of h)
we find: .

However the sad part, your expression can not be simplified this way
Then you have to stick to kompiks method.
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