ShmoopTube

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Here's another shmoop du jour..

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Which of the following is equal to the root

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3 minus 1 over root 3 plus root 5? And here are your potential answers...

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Okay, any time we get one of these, it's a good bet that we want to multiply by some

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combination of the denominator to hopefully just simplify it.

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So in this case - we want a perfect square that's in the form of (a +b) times (a - b)...

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Since we have the square root of 3 plus the square root of 5 on the bottom...

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we have to multiply the fraction

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by root 3 minus root 5 times over root 3 minus root 5...which is the equivalent of 1.

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Based on the perfect square formula... with a equaling root 3 and b equaling

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root 5...we have the bottom of the fraction simplifying to 3 minus 5...or negative 2.

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We can use simple distribution properties for the top of the fraction.

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Root 3 times root 3 equals 3. Root 3 times negative root 5 equals negative root 15...negative

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1 times root 3 equals negative root 3...and negative 1 times negative root 5 equals positive root 5.

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All over NEGATIVE 2.

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But looking at our answer choices...none of them have a negative 2 at the bottom.

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So let's just move it to the top. Keep in mind that we have to take the negative of

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the ENTIRE top of the fraction...so we have to distribute a negative to each number...

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Negative 3 plus root 15 plus root 3 minus root 5--all over 2.

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Looks like answer choice E is the right one.

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