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Help..! Roots Q

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by rmjmcc » Mon Oct 28, 2013 10:05 am
Hello
I have been going through my notes and came across the following question, which I just cannot seem to solve. I assume I am missing a very obvious step?

1/ sqrt(29) + sqrt(20)

Unfortunately I can't remember where this came from, so I don't have the A-E proposed solutions either...
Sorry!
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by theCodeToGMAT » Mon Oct 28, 2013 10:10 am
Do you mean

[1/sqrt(29)] + sqrt(20)

or

1/[sqrt(29) + sqrt(20)]
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by rmjmcc » Mon Oct 28, 2013 10:20 am
The latter:

1/((sqrt(29) + sqrt(20))
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by Matt@VeritasPrep » Mon Oct 28, 2013 10:28 am
If you have √29 + √20 and you want to get rid of the roots without adding new terms, a nice trick is to multiply the expression by √29 - √20. Watch what happens:

(√29 + √20) * (√29 - √20)
= 29 + √29√20 - √29√20 - 20
= 9

Sweet!

Since you can't actually change the value of your fraction, to change the expression you'll multiply both the top AND the bottom by this number:

1/(√29 + √20) * (√29 - √20)/(√29 - √20)

which gives you

(√29 - √20)/9

And you're done!
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by Brent@GMATPrepNow » Mon Oct 28, 2013 10:39 am
Matt has done a great job demonstrating that 1/(√29 + √20) is equivalent to (√29 - √20)/9, but you may be wondering why we'd bother doing this since it doesn't exactly simplify matters. Here's the reason:

In mathematics, it's generally considered poor form to write fractions with roots in the denominator (although having roots in the numerator is acceptable). So, to eliminate the roots in the denominator, we must multiple both numerator and denominator by something that accomplishes this. Here that something is (√29 - √20).

Aside: √29 - √20 is known as the "conjugate" of √29 + √20

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Brent
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by rmjmcc » Mon Oct 28, 2013 11:01 am
Thank you!
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by GMATGuruNY » Mon Oct 28, 2013 11:26 am
rmjmcc wrote:Hello
I have been going through my notes and came across the following question, which I just cannot seem to solve. I assume I am missing a very obvious step?

1/ sqrt(29) + sqrt(20)

Unfortunately I can't remember where this came from, so I don't have the A-E proposed solutions either...
Sorry!
The GMAT would provide answer choices.
If the answer choices are sufficiently spread out, we can quickly determine the correct answer choice by BALLPARKING.
√25 = 5.
Thus:
1/(√29 + √20) = 1/(a bit more than 5 + a bit less than 5) ≈ 1/10.
The correct answer choice must be close to 1/10.
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