Simplify the following rational numbers:

by

Simplify:

[1/(√4 + √5)] + [1/(√5 + √6)] + [1/(√6 + √7)] + [1/(√7 + √8)] + [1/(√8 + √9)]

Solution:

Let us simplify individually,
[1/(√4 + √5)]
Rationalize the denominator, we get
[1/(√4 + √5)] = [1(√4 – √5)] / [(√4 + √5) (√4 – √5)]
= [(√4 – √5)] / [(√4)2 – (√5)2]
= [(√4 – √5)] / [4 – 5]
= [(√4 – √5)] / -1
= -(√4 – √5)
Now,
[1/(√5 + √6)]
Rationalize the denominator, we get
[1/(√5 + √6)] = [1(√5 – √6)] / [(√5 + √6) (√5 – √6)]
= [(√5 – √6)] / [(√5)2 – (√6)2]
= [(√5 – √6)] / [5 – 6]
= [(√5 – √6)] / -1
= -(√5 – √6)
Now,
[1/(√6 + √7)]
Rationalize the denominator, we get
[1/(√6 + √7)] = [1(√6 – √7)] / [(√6 + √7) (√6 – √7)]
= [(√6 – √7)] / [(√6)2 – (√7)2]
= [(√6 – √7)] / [6 – 7]
= [(√6 – √7)] / -1
= -(√6 – √7)
Now,
[1/(√7 + √8)]
Rationalize the denominator, we get
[1/(√7 + √8)] = [1(√7 – √8)] / [(√7 + √8) (√7 – √8)]
= [(√7 – √8)] / [(√7)2 – (√8)2]
= [(√7 – √8)] / [7 – 8]
= [(√7 – √8)] / -1
= -(√7 – √8)
Now,
[1/(√8 + √9)]
Rationalize the denominator, we get
[1/(√8 + √9)] = [1(√8 – √9)] / [(√8 + √9) (√8 – √9)]
= [(√8 – √9)] / [(√8)2 – (√9)2]
= [(√8 – √9)] / [8 – 9]
= [(√8 – √9)] / -1
= -(√8 – √9)
So, according to the question let us substitute the obtained values,
[1/(√4 + √5)] + [1/(√5 + √6)] + [1/(√6 + √7)] + [1/(√7 + √8)] + [1/(√8 + √9)]
= -(√4 – √5) + -(√5 – √6) + -(√6 – √7) + -(√7 – √8) + -(√8 – √9)
= -√4 + √5 – √5 + √6 – √6 + √7 – √7 + √8 – √8 + √9
= -√4 + √9
= -2 + 3
= 1

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