Hello,
Answer:
We can start by rationalizing the denominators of each fraction.
For the first fraction, we multiply the numerator and denominator by the conjugate of the denominator, which is √4 - √5:
1/√4+√5 = 1/√4+√5 × (√4 - √5)/(√4 - √5)
= (√4 - √5)/(4 - 5)
= (√4 - √5)/(-1)
For the second fraction, we multiply the numerator and denominator by √5 - √6:
1/√5+√6 = 1/√5+√6 × (√5 - √6)/(√5 - √6)
= (√5 - √6)/(5 - 6)
= (√5 - √6)/(-1)
We can do the same for the remaining fractions:
1/√6+√7 = (√6 - √7)/(-1)
1/√7+√8 = (√7 - √8)/(-1)
1/√8+√9 = (√8 - √9)/(-1)
Now we can simplify the expression:
1/√4+√5 + 1/√5+√6 + 1/√6+√7 + 1/√7+√8 + 1/√8+√9
= (√4 - √5)/(-1) + (√5 - √6)/(-1) + (√6 - √7)/(-1) + (√7 - √8)/(-1) + (√8 - √9)/(-1)
= (√4 - √5 + √5 - √6 + √6 - √7 + √7 - √8 + √8 - √9)/(-1)
= (√4 - √9)/(-1)
= √9 - √4
= 3 - 2
= 1
Therefore, the expression simplifies to 1, which is equal to the right-hand side of the equation.
Good luck