How do you simplify -1*sqrt(2)*sqrt(2-sqrt(3)) −1⋅√2⋅√2−√3?

1 Answer
Zack M.
Oct 20, 2015

1-sqrt(3)1−√3

Explanation:

The rules for radicals say that we can multiply the terms of the same order together, so we can move the 22 under the second radical term to get;

-1*sqrt(2(2-sqrt(3))−1⋅√2(2−√3)

Multiplying the 22 through, we get;

-sqrt(4-2sqrt(3))−√4−2√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)√(x+y)−2√xy=√x−√y

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

4=(3+1)4=(3+1)
3=3*13=3⋅1

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

-sqrt((3+1) -2sqrt(3*1)) −√(3+1)−2√3⋅1

= -(sqrt(3)-sqrt(1))=−(√3−√1)

=1-sqrt(3)=1−√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)−2√xy)2=(√x−√y)2

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

x-2sqrt(xy) +y = x-2sqrt(xy) + yx−2√xy+y=x−2√xy+y

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