How do you simplify #-1*sqrt(2)*sqrt(2-sqrt(3)) #?

1 Answer
Oct 20, 2015

Answer:

#1-sqrt(3)#

Explanation:

The rules for radicals say that we can multiply the terms of the same order together, so we can move the #2# under the second radical term to get;

#-1*sqrt(2(2-sqrt(3))#

Multiplying the #2# through, we get;

#-sqrt(4-2sqrt(3))#

A Google search on simplifying nested radicals turned up this page, which tells us that we can simplify this expression a little further using the rule;

#sqrt((x+y)-2sqrt(xy))=sqrt(x) - sqrt(y)#

Our expression is certainly the right form, but we need to check that we have reasonable #x# and #y# values to plug in. It turns out that;

#4=(3+1)#
#3=3*1#

Therefore, we can use #x=3# and #y=1# to simplify our expression.

#-sqrt((3+1) -2sqrt(3*1)) #

#= -(sqrt(3)-sqrt(1))#

#=1-sqrt(3)#

*Note: the page referenced above does not provide any proof for the equation mentioned, but if you square both sides, you can see that they are indeed equal.

#(sqrt((x+y)-2sqrt(xy)))^2=(sqrtx - sqrty)^2#

#(x+y)-2sqrt(xy)=(sqrtx-sqrty)(sqrtx-sqrty)#

#x-2sqrt(xy) +y = x-2sqrt(xy) + y#