1 9- 7- 1 7- 5+ 1 5- 3- 1 3- 1=

Free Practice Question

PS · Problem Solving

655-705

Roots

$\frac{1}{\sqrt{9}-\sqrt{7}}-\frac{1}{\sqrt{7}-\sqrt{5}}+\frac{1}{\sqrt{5}-\sqrt{3}}-\frac{1}{\sqrt{3}-\sqrt{1}}=$

Correct answer marked below

\frac{1}{2}

Correct

2\sqrt{2}

4-\sqrt{7}+\sqrt{5}-\sqrt{3}

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Explanation

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Comments

FrancescoJul 13

Rationalizing each term one by one took a while but eventually the pattern showed up. Definitely not a 30-second question for me.

JotiJul 22

Same here. I almost gave up before noticing the signs start alternating after rationalizing.

AtishayJul 27

How did you keep track of the signs? I kept messing up the third term.

OmegaAug 19

Is there a faster way than rationalizing all four denominators? Feels like the intended trick is something with conjugates but I'm not seeing it yet.

TomAug 31

Multiply numerator and denominator by the conjugate for each term, then just watch what cancels.

SherveerApr 18

Got a nice telescoping after rationalizing, very satisfying once it clicked.

TafreeMay 2

Yeah, the middle terms cancel out neatly. Took me two attempts to see it though.

ManaswiAug 1

The answer choices with radicals in them are tempting if you don't simplify fully. Classic trap.

What this question tests

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

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