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[(5√3+3√5)/(√5+√3)]2=?

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[(5√3+3√5)/(√5+√3)]2=?

by Max@Math Revolution » Wed Sep 28, 2016 3:03 am
[(5√3+3√5)/(√5+√3)]2=?
A. √6 B. √15 C. 15 D. √24 E. 6
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by Brent@GMATPrepNow » Wed Sep 28, 2016 6:58 am
Max@Math Revolution wrote:[(5√3 + 3√5)/(√5 + √3)]² = ?
A. √6
B. √15
C. 15
D. √24
E. 6
[(5√3 + 3√5)/(√5 + √3)]²
Square numerator and denominator to get: [75 + 15√15 + 15√15 + 45]/[5 + √15 + √15 + 3]
Simplify to get: [120 + 30√15]/[8 + 2√15]

NOTE: At this point, we COULD multiply top and bottom by the conjugate (8 - 2√15). HOWEVER, we can save time by recognizing that we can first factor the top and bottom.

Factor the top and bottom to get: [30(4 + √15)]/[2(4 + √15)]
Eliminate (4 + √15) from top and bottom to get: 30/2
Evaluate: 15

Answer: D
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by GMATGuruNY » Wed Sep 28, 2016 7:48 am
Max@Math Revolution wrote:[(5√3+3√5)/(√5+√3)]²=?
A. √6 B. √15 C. 15 D. √24 E. 6
An alternate approach is to BALLPARK.

Every test-taker should know the values of common roots.
√3 ≈ 1.7.
√5 ≈ 2.3.

Thus:
[(5√3+3√5)/(√5+√3)]²

≈ [[(5)(1.7) + (3)(2.3)]/(2.3 + 1.7)]²

≈ [(8.5 + 7)]/4]²

≈ (16/4)²

= 4²

= 16.

Only C is close to 16.

The correct answer is C.
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by Max@Math Revolution » Fri Sep 30, 2016 6:44 am
==> [(5√3+3√5)/(√5+√3)]2=[√15(√5+√3)/(√5+√3)]2=(√15)2=15 Hence the answer is C.
Answer: C
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