√(3+2√(2)) - √(3-2√(2))

I used a calculator to get the answer 2. How would I show that the calculated value is correct?

Question

√(3+2√(2)) - √(3-2√(2))

I used a calculator to get the answer 2. How would I show that the calculated value is correct?

amistre64 · Answer

by another calculator?

amistre64 · Answer

you have a smaller number being subtracted by a bigger number, so that answer is gonna be +

amistre64 · Answer

equate this this to some value "N" and solve it algebraiczlly

amistre64 · Answer

typoed it :)

you have a smaller number being subtracted FROM a bigger number, so that answer is gonna be +

anonymous · Answer

but how would I prove that the answer is 2, rather than any other positive integer?

amistre64 · Answer

by solving for "N" of course.

$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}=N$ what do you think we can do with this?

amistre64 · Answer

or rather, how do we undo square roots?

anonymous · Answer

square both sides?

amistre64 · Answer

id say thats a good start :)

$\left(\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}\right)^2=N^2$

now we can simplify this :)

amistre64 · Answer

(a-b)^2 = (a^2 -2ab + b^2) if that helps

amistre64 · Answer

$[3+2\sqrt{2}]-[2\sqrt{(3+2\sqrt{2})(3-2\sqrt{2})}]+[3-2\sqrt{2}]=n^2$

amistre64 · Answer

$[3+2\sqrt{2}]+[3-2\sqrt{2}]-[2\sqrt{(3+2\sqrt{2})(3-2\sqrt{2})}]=n^2$

$6-2\sqrt{9-8}=n^2$

$6-2\sqrt{1}=n^2$

$6-2=n^2$

$4=n^2=2$

amistre64 · Answer

well; N=2 it i can type :)

anonymous · Answer

Thank you so much :)