How do you simplify square root of 30 - square root of 3?

1 Answer
George C.
Jul 3, 2018

Both #sqrt(30)-sqrt(3)# and #sqrt(30-sqrt(3))# are in simplest form already.

Explanation:

It is not clear from the question whether you intend:

#sqrt(30)-sqrt(3)#

or:

#sqrt(30-sqrt(3))#

Note that:

#30 = 2 * 3 * 5#

has no square factors, so cannot be simplified.

However, note that one of its factors is #3#. So it is possible to split #sqrt(30)# into a product of square roots that we can group as follows:

#sqrt(30)-sqrt(3) = sqrt(10) * sqrt(3) - 1 * sqrt(3) = (sqrt(10)-1)sqrt(3)#

I am not sure that you would call that a simplification, but it might be a useful alternative form.

In the case of #sqrt(30-sqrt(3))# we might try looking for an expression of the form #a+bsqrt(3)# whose square is #30-sqrt(3)#, but this does not lead to a simplification...

#(a+bsqrt(3))^2 = (a^2+3b^2)+(2ab)sqrt(3)#

So:

#{ (a^2+3b^2 = 30), (2ab = -1) :}#

Putting #b = -1/(2a)# the first equation becomes:

#a^2+3/(4a^2)=30#

and hence:

#a^4-30a^2+3/4 = 0#

and hence:

#4a^4-120a^2+3 = 0#

from which we find:

#a^2 = 15+-sqrt(897)/2#

So there are no nice rational or simple irrational values of #a, b# such that #(a+bsqrt(3))^2 = sqrt(30-sqrt(3))#

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