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anuu
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Exponent problem Help! 08 May 2011, 21:38
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Hi guys,

I came across the below problem in an online test. any suggestions pls to solve the problem?

if (7+4SQRT3)^x2-8 + (7-4SQRT3)^x2-8 = 14, what is x?


The source of hte problem is the following URL:

https://gmat.jumbotests.com/tests/gmat- ... -equations

The exponent part in both the terms is x^2-8

(Question 1 of 10)

Regards,
Anu



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Exponent problem Help! 08 May 2011, 22:04
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i will go with +-3, +- sqrt 7

its easier to fix 3 and -3 in the eqn and see, this will eliminate 2 choices. And then try -1 will eliminate another one choice. And then try -sqrt(7) to get the answer.

Hope it helps.
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Exponent problem Help! 08 May 2011, 22:08
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It is x^2, hence 3| -3 is a solution.

Also, 7^(1/2) | - 7^(1/2) is also a solution too.

B.

for 7^(1/2), the equation becomes 1/(7 + 4* 3^1/2) + 1/(7 - 4* 3^1/2) = 14 / (49-48) = 14.
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anuu
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Exponent problem Help! 08 May 2011, 22:32
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agdimple333
i will go with +-3, +- sqrt 7

its easier to fix 3 and -3 in the eqn and see, this will eliminate 2 choices. And then try -1 will eliminate another one choice. And then try -sqrt(7) to get the answer.

Hope it helps.
Show more


Thank you!! for the quick reply. this helps.

The only way to solve such problems is to directly plug in the values is it?Is there any approach? in some exponent problems similar to this, we use multiple formulas like (a+b)^2 and then (a+b)(a-b)...

Anuu
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Exponent problem Help! 08 May 2011, 23:11
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I'll go with a plug and play approach. But I'll make life easier with some hints like the following:

The moment I see a surd, I first try to see what its reciprocal is and then look for what its conjugate is. A surd is an expression of the form a + b*sqrt(c) where a, b and c are rational numbers. Its reciprocal is 1/(a + b*sqrt(c)); and its conjugate is a - b*sqrt(c).

One interesting coincidence (by design) in most of these GMAT questions is that the reciprocal and the conjugate are equal. In other words, the following equation holds for most problems.

a - b*sqrt(c) = 1/(a+b*sqrt(c)).

Observe that this question is no different.

The moment you see this relation, you have the first breakthrough for the answer.

Plugging 3 into the expression is no difficult task. But understanding the sqrt(7) part will be extremely faster if you bear the above generalization in mind.

Now if the RHS is a rational number, I look for an addition operation between the surd and its conjugate. If the RHS contained a surd too, I'd be looking to subtract the main surd from its conjugate (in the above surd, it becomes (+/-) 2*b*sqrt(c).

My strong learnings and leanings when I attack GMAT math: Do not become too mathematical when trying to solve any question. If necessary, unlearn all higher grade mathematics.

Regards
Rahul
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Exponent problem Help! 08 May 2011, 23:16
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By the way, I'd not call this an exponent problem. It is a surd problem. Elders are free to disagree ;-)

Regards
Rahul



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