Simplifying a Nice Radical Expression in Two Ways

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#ChallengingMathProblems #RadicalExpressions

PLAYLISTS 🎵 :

[田村博志-zy]

To simplify the notation we define
a := √( 2 - √3 ),
b := √( 2 + √3 ).
Then we have 0 < a < b and
a^2 = 2 - √3,
b^2 = 2 + √3,
a^2 + b^2 = 4,
ab = 1,
( b - a )^2 = a^2 - 2ab + b^2 = 2,
b - a = √2.
We write the expression as
b^2/( √2 + b ) + a^2/( √2 - a ) = ( a^2( √2 + b ) + b^2( √2 - a ) )/( ( √2 - a )( √2 + b ) )
= ( √2( a^2 + b^2 ) - ab( b - a ) )/( 2 - ab + √2( b - a ) )
= ( 4√2 - √2 )/( 2 - 1 + 2 )
= √2.
The answer is √2.

[joyli]

WOW! You teach us NEW MATH every week! When we learn more, we get smarter.

[kamaljain]

you can also multiple both numerator and denominator with \sqrt(2). if you do that, \sqrt(2) in the denominator makes \sqrt(2+\sqrt(3)) term into \sqrt(4 + 2\sqrt(3)) = \sqrt(3+ 2\sqrt(3) + 1) = 1 + \sqrt(3). same happens for \sqrt(2-\sqrt(3)) term. everything in the numerator and denominator cancels out, and you are left with \sqrt(2) you multiplied in the numerator.

[gulzarsingh]

I put the given expression = x
First i found (√2)x = [2/√2]*x
In 1st fraction, numerator becomes [4 + 2√3] = [ (√3) + 1]^2
& denominator becomes 2 + sqrt[4 + 2√3)] = 2 + [(√3) + 1] = 3 + (√3) = √3[(√3) + 1]
On simplifying the fraction became [(√3)+1] / √3
By similar process, the second fraction becomes [(√3) – 1] / √3
By adding, it becomes (2√3)/(√3) = 2
(√2)x = 2 ⇒ x = √2

[hornet-eg]

awesome! i wonder at how much work you put into these videos keep it up😄

[songuiuc]

You can also add the two fractions using a common denominator. There will be a lot of difference of squares.
(2+√3)* √(2-√3)=

[mymathprogress]

Those two were nice ways to handle this problem. However, I like the second one as it appears to be a simpler solution to the problem.

[aweebthatlovesmath]

I can't believe that mess was just √2 🤯

[Beathan]

I wonder what kind of software is used to create such a nice lessons?

[victorabaderamos]

suggestion for a future video: cos(1/x)=sin(x)

[نمبروان-عم]

I dont understand can you please explain slowly ???!!

[srividhyamoorthy]

Nice video I mean its going to be nice

[voltalimwabbit]

You were supposed to explain it to your viewers and should have done it more slowly. You sounded to be such in a great rush this time. A bit disappointed.