Question

Simplify:
1/√4+√5 + 1/√5+√6 + 1/√6+√7 + 1/√7+√8 + 1/√8+√9

Answer 1

1

Explanation:

Simplfy the radicals

Identity used (a - b)(a+b)= a² -b²

• Rationalize the deniminator and to do that multiply the denominator and numerator by the conjugate of the denominator.

`\text{Conjugate of $√(5)+√(4) = √(5)-√(4)$}`

`\sf (1)/(√(5)+√(4))=(1*(√(5)-√(4)))/((√(5)+√(4))(√(5)-√(4)))`

`\sf =(√(5)-√(4))/((√(5))^2-(√(4))^2)\\ \\ = (√(5) -√(4))/(5-4)\\\\ = (√(5)-√(2*2))/(1)\\\\= √(5)-2`

In the same way,

`(1)/(√(5)+√(6))=√(6)-√(5)\\\\(1)/(√(7)+√(6))=√(7)-√(6)\\\\(1)/(√(8)+√(7))=√(8)-√(7)\\\\(1)/(√(9)+√(8))=√(9)-√(8)= √(3*3)-√(8)=3-√(8)`

`(1)/(√(4)+√(5))+(1)/(√(5)+√(6))+(1)/(√(6)+√(7))+(1)/(√(7)+√(8))+(1)/(√(8)+√(9)) \\\\\\=√(5)-2+√(6)-√(5) +√(7)-√(6)+√(8)-√(7)+ 3 -√(8)`

= -2 + 3

= 1