What is the value of 99+ 33 99- 33+ 99- 33 99+ 33?

Free Practice Question

PS · Problem Solving

555-605

Roots

What is the value of $\frac{\sqrt{99}+\sqrt{33}}{\sqrt{99}-\sqrt{33}}+\frac{\sqrt{99}-\sqrt{33}}{\sqrt{99}+\sqrt{33}}$ ?

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Comments

ElouanApr 25

Thanks, the symmetry of the two fractions made this click for me.

SaurabhMay 8

same, once I noticed they were reciprocals it was quick

KunMay 8

I almost expanded everything at first lol

JaseemMay 1

Is it fair to let x be the first fraction and notice the second is 1/x? That seems cleaner than rationalizing twice.

AnuragMay 14

Yep, I did that and it saved a ton of time.

TanishkaSep 25

I got a bit lost with the radicals. Do you combine the fractions over a common denominator first, or use the difference of squares trick?

TriOct 1

Try letting a = sqrt(99) and b = sqrt(33), then write it as (a+b)^2+(a-b)^2 over (a+b)(a-b).

DibyashreeJun 16

This one felt very doable for 555-605, the pattern basically gives it away.

AngelinaJun 26

Agreed, once you see it's a reciprocal pair it's not bad at all.

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.

What is the value of 99+3399−33+99−3399+33\frac{\sqrt{99}+\sqrt{33}}{\sqrt{99}-\sqrt{33}}+\frac{\sqrt{99}-\sqrt{33}}{\sqrt{99}+\sqrt{33}}99​−33​99​+33​​+99​+33​99​−33​​?