If x = 1 11+ 10 and y = 1 11- 10, then what is the value of x^2 - xy + y^2?

Free Practice Question

PS · Problem Solving

605-655

Roots

If $x = \frac{1}{\sqrt{11}+\sqrt{10}}$ and $y = \frac{1}{\sqrt{11}-\sqrt{10}}$ , then what is the value of $x^2 - xy + y^2$ ?

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47

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Comments

CoverMay 14

Rationalizing both denominators makes this way less scary. I almost tried to plug into the expression directly and got lost in radicals lol.

AnjelinaMay 16

Same here, then I realized x and y become conjugates after rationalizing. Much cleaner.

GaganMay 18

Yeah once you see x = sqrt11 - sqrt10 and y = sqrt11 + sqrt10 it's pretty smooth.

PoojaJul 17

Is there a shortcut using (x - y)^2 + xy? I keep messing up the middle term sign when I square things.

DhruvJul 22

Yep, x^2 - xy + y^2 = (x-y)^2 + xy. That saved me from expanding everything.

PaulJul 31

Just be careful: x - y will be negative, so squaring it is fine but xy is also negative.

FrancescoMay 30

For me the trickiest part was xy after rationalizing. x and y multiply to 1 because the denominators are conjugates, right?

ArinJun 3

Yes, (sqrt11 - sqrt10)(sqrt11 + sqrt10) = 1. That's the key step.

ClifordMay 11

Good 600-level question. Not too hard if you know conjugates, but rationalizing both fractions first is essential. Took me about 1:30.

What this question tests

What it tests

Your handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.

If x=111+10x = \frac{1}{\sqrt{11}+\sqrt{10}}x=11​+10​1​ and y=111−10y = \frac{1}{\sqrt{11}-\sqrt{10}}y=11​−10​1​, then what is the value of x2−xy+y2x^2 - xy + y^2x2−xy+y2?