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What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4

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MathRevolution
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What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4 [#permalink] New post  05 Apr 2019, 01:43
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[GMAT math practice question]

What is the simplified expression for \(\frac{1}{(√1+√2)} + \frac{1}{(√2+√3)} + \frac{1}{(√3+√4)} + ... + \frac{1}{(√99+√100)}?\)

\(A. 8\)
\(B. 9\)
\(C. 10\)
\(D. 11\)
\(E. 12\)
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What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4 [#permalink] New post  05 Apr 2019, 02:41
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MathRevolution wrote:
[GMAT math practice question]

What is the simplified expression for \(\frac{1}{(√1+√2)} + \frac{1}{(√2+√3)} + \frac{1}{(√3+√4)} + ... + \frac{1}{(√99+√100)}?\)

\(A. 8\)
\(B. 9\)
\(C. 10\)
\(D. 11\)
\(E. 12\)


Rationalizing denominators of each of the irrational fractions by respective conjugate pairs, we have
\(\frac{(√2-√1)}{(√2+√1)(√2-√1)} +\frac{(√3-√2)}{(√3+√2)(√3-√2)}+---------+\frac{(√100-√99)}{(√100+√99)(√100-√99)}\)
=\(\frac{(√2-√1)}{(2-1)}+ \frac{(√3-√2)}{(3-2)}+---------+\frac{(√100-√99)}{(100-99)}\)
=√2-√1+√3-√2+................+√100-√99
=-1+10
=9

Ans. (B)
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Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4 [#permalink] New post  05 Apr 2019, 02:57
Ans. B

Rationalize the denominator
End up with -9/-1 = 9

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MathRevolution
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Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4 [#permalink] New post  07 Apr 2019, 18:43
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=>

\(\frac{1}{(√1+√2)} + \frac{1}{(√2+√3)} + \frac{1}{(√3+√4)} + ... + \frac{1}{(√99+√100)}\)

\(=\frac{1}{(√2+√1)} + \frac{1}{(√3+√2)} + \frac{1}{(√4+√3)} + ... + \frac{1}{(√100+√99)}\)

\(= \frac{(√2-√1)}{{(√2+√1)(√2-√1)}} + \frac{(√3-√2)}{{(√3+√2)(√3-√2)}} + \frac{(√4-√3)}{{(√4+√3)(√4-√3)}} + ... + \frac{(√100-√99)}{{(√100+√99)(√100-√99)}}\)

\(= (√2-√1) + (√3-√2) + (√4-√3) + ... + (√100-√99)\)

\(= (-√1+√2) + (-√2+√3) + (-√3+√4) + ... + (-√99+√100)\)

\(= -1+10\)

\(= 9\)

Therefore, the answer is B.
Answer: B
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Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4 [#permalink] New post  03 Sep 2021, 02:29
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Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4 [#permalink]
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