Question

Rationalize the denominator and simplify.

Answer 1

`=(a+5-4√(a+1))/(a-3)`

Explanation:

Given:

`(√(a+1)-2)/(√(a+1)+2)`

Rationalise the denominator.

Solution:

Simplify the expression.

`=(√(a+1) -2)/(√(a+1) +2 )`

Multiply numerator by both denominator and numerator.

`=(√(a+1) -2)/(√(a+1)+2)* (√(a+1)-2)/(√(a+1)-2)`

`=((√(a+1)-2)(√(a+1)-2))/((√(a+1)+2)(√(a+1)-2))`

Assume
`√(a+1) =a\ and\ 2=b`

Applying formula
`(a-b)(a+b)=a^(2) -b^(2)`, so we get

`=((√(a+1) -2)^(2))/((√(a+1))^(2) -2^(2)))`

`=((√(a+1))^(2) +2^(2)-2(√(a+1))(2))/((a+1)-4)`

`=((a+1) +4-4√(a+1))/(a+1-4)`

`=(a+5-4√(a+1))/(a-3)`

Therefore, the simplification of the expression is given below.

`=(a+5-4√(a+1))/(a-3)`