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 Post subject: GMAT ArithmeticPosted: Tue Aug 17, 2010 11:26 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
(√54 – √6) / (√54 + √6) =
A. 1/3
B. √6/6
C. 1/2
D. 5/8
E. 1

(C) √54 = √(9 × 6) = 3√6.

So (√54 – √6) / (√54 + √6) = (3√6 – √6) / (3√6 + √6) = (2√6) / (4 √6) = 1/2.

ALTERNATIVE SOLUTION:
When we see expression in denominator that includes radicals the first method to simplify it is to use formula a² – b² = (a + b)(ab). We see that in this case we have only (a + b) in the denominator. So let us multiply both numerator and denominator by (√54 – √6):
[(√54 – √6) × (√54 – √6)] / [(√54 + √6) × (√54 – √6)] =
= (√54 – √6)² / (54 – 6) = (54 + 6 – 2 × 18) / 48 = 24 / 48 = 1/2

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Hello. I do not understand how the numerator breaks out into (54 + 6 – 2 × 18). Please explain. Thank you.

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 Post subject: Re: math (test 1, question 8): fractions, irrationalityPosted: Tue Aug 17, 2010 11:30 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
When we multiply both numerator and denominator by (√54 – √6) numerator becomes:
(√54 – √6) × (√54 – √6) = (√54 – √6)² =
= (√54)² + (√6)² – 2 × √54 × √6 =
= 54 + 6 – 2 × √(54 × 6) = 54 + 6 – 2 × 18 = 24

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 Post subject: Re: math (test 1, question 8): fractions, irrationalityPosted: Tue Nov 02, 2010 1:45 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
Where did the 18 come from please?

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 Post subject: Re: math (test 1, question 8): fractions, irrationalityPosted: Tue Nov 02, 2010 1:51 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
18 is √(54 × 6)

√(54 × 6) = √(2 × 3 × 3 × 3 × 2 × 3) = √(2 × 3 × 3)² = √(18²) = 18

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