What is the value of \left( 1 5+ 3+ 1 5- 3\right)^2?

Free Practice Question

PS · Problem Solving

555-605

Roots

What is the value of $\left(\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{5}-\sqrt{3}}\right)^2$ ?

Correct answer marked below

\frac{1}{16}

\frac{1}{4}+\sqrt{15}

1

5

Correct

9

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Explanation

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Comments

JacobJun 9

Rationalizing the denominators made this pretty quick for me. Nice one.

AigulJun 15

Yeah once you see both denominators turn into 2 it's smooth sailing.

DhruvApr 21

Wait, do I have to rationalize both fractions separately? I tried combining them over a common denominator first and got a mess.

AlexMay 1

Combining first works too but you'll deal with (a+b)(a-b) anyway.

Filipe1d ago

Got 5 as my answer. The two terms end up being conjugates of each other so they add nicely.

AnubhavMay 8

Is this really 555-605? Took me a bit to remember the conjugate trick but after that it's easy.

HarshMay 14

I'd say yes, it's mostly testing whether you spot the rationalization.

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.