Questions :
Evaluate the expression: (1/(√10 − √9)) − (1/(√9 − √8)) + (1/(√8 − √7)) − (1/(√7 − √6)) + (1/(√6 − √5)) − (1/(√5 − 2)) = ?
Options :
- √10 − 2
- 8
- 2
- √10 + 2
correct answer : a)
Explanation :
Rationalize each term using:
1/(√a − √b) = (√a + √b)/(a − b) = √a + √b
Therefore,
1/(√10 − √9) = √10 + √9
1/(√9 − √8) = √9 + √8
1/(√8 − √7) = √8 + √7
1/(√7 − √6) = √7 + √6
1/(√6 − √5) = √6 + √5
1/(√5 − 2) = 1/(√5 − √4) = √5 + 2
Substituting:
(√10 + √9) − (√9 + √8) + (√8 + √7) − (√7 + √6) + (√6 + √5) − (√5 + 2)
All middle terms cancel out:
= √10 − 2
Therefore, the value of the expression is √10 − 2. Hence, the correct answer is option (a). The values 8, 2, and √10 + 2 do not match the simplified result and are therefore incorrect.