Which of the following is equal to 8-4 3+ 8+4 3?

Free Practice Question

PS · Problem Solving

655-705

Roots

Which of the following is equal to $\sqrt{8-4\sqrt{3}}+\sqrt{8+4\sqrt{3}}$ ?

Correct answer marked below

2\sqrt{3}

2\sqrt{5}

2\sqrt{6}

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Explanation

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Comments

HuanJul 16

This one took me way too long. I kept trying to square the whole thing but got lost in the algebra. Is the trick to rewrite 8±4√3 as perfect squares?

DevanshJul 25

Yeah exactly, think of (a±b)^2 form. That's what unlocked it for me.

PatJul 23

Took me a sec too, but once you see the pattern it's fast.

AndreaSep 20

I initially picked 2√6 because I just added the inside terms like an idiot. Lesson learned: always check if the radicals simplify first.

AshitaSep 21

Same trap here. The inner terms look like they should add nicely but they don't.

KshitizSep 22

The key is noticing 8 - 4√3 = (√6 - √2)^2 and 8 + 4√3 = (√6 + √2)^2. Then it collapses beautifully.

Parth6d ago

Nice, that's clean. I was trying to square the whole expression instead.

WarisaMay 21

Really good question. 655-705 feels right, it's not hard conceptually but easy to mess up under time pressure.

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.