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

2 Answers

We have that

sqrt(3-2*sqrt(2))=sqrt((sqrt2)^2-2sqrt2*1+1^2)=(sqrt((sqrt2-1))^2)=sqrt2-1

Jun 27, 2018

Take the needed answer form with variables and solve for them

Explanation:

The answer above is completely correct, but it may help the student to see how to reach it:

We seek a square root of 3-2sqrt(2). This must have the form a+bsqrt(2).

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

So
a^2+2b^2=3
and
2ab=-2rArrab=-1
These are two simultaneous equations for a and b.

Rearrange the second:
b=-1/a
Substitute into the first:
a^2+2/a^2=3
a^4+2=3a^2
a^4-3a^2+2=0

Note that this is a quadratic in a^2:
(a^2)^2-3(a^2)+2=0
Factorise:
(a^2-2)(a^2-1)=0

This gives us two possible solutions for a^2: 2 and 1, and so the four solutions for a: +-sqrt(2) and +-1.

We are looking for integer solutions for a, and so +-1 are possible solutions. But the other two are possible too - they can simply be folded in to the sqrt(2) term. This wouldn't have been possible if we'd had the root of some other number in the solution for a, but this solution is a special case.

Now use the second equation to deduce the four equivalent solutions for b:
b=-1/a
b=bar(+)1/sqrt(2)=bar(+)1/2sqrt(2) and bar(+)1.

So we have the four solution pairs (a,b):
(sqrt(2),-1/2sqrt(2))
(-sqrt(2),1/2sqrt(2))
(1,-1)
(-1,1)

This is a bit suspicious - we expect only two solutions, positive and negative square roots, so we wonder if some of these are identical to each other: When we substitute them in to our desired expression a+bsqrt(2), we get:

sqrt(2)-1/2sqrt(2)sqrt(2)=-1+sqrt(2)
-sqrt(2)+1/2sqrt(2)sqrt(2)=1-sqrt(2)
1-sqrt(2)
-1+sqrt(2)

So the two solutions with sqrt(2) are identical to the two simpler solutions, so we can get rid of them. We now have two solutions, positive and negative square roots:
1-sqrt(2)
-1+sqrt(2)

When we take the written square root of a quantity, it is implied that the desired root is the positive root, the "principal value" of the square root function. So we take the single solution that comes from a=+1:
1-sqrt(2)

Double check: Make sure that this produces the desired answer:
(1-sqrt(2))^2=1-2sqrt(2)+2=3-2sqrt(2)