Question

Express each one of the following with rational denominator:

`(b^2)/(sqrt(a^2 + b^2) + a)`

Answer 1

We know that rationalization factor for `sqrt(a^2 + b^2) + a` is `sqrt(a^2 + b^2) - a`. We will multiply numerator and denominator of the given expression `b^2/(sqrt(a^2 + b^2) + a) ` by `sqrt(a^2 + b^2) - a` to get

`b^2/(sqrt(a^2 + b^2) + a) xx (sqrt(a^2 + b^2) - a)/(sqrt(a^2 + b^2) - a) = (b^2(sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2) - a^2)`

`= (b^2 (sqrt(a^2 + b^2) - a))/(a^2 + b^2 - a^2)`

`= (b^2(sqrt(a^2 + b^2) - a))/b^2`

`= sqrt(a^2 + b^2) - a`

Hence the given expression is simplified with rational denominator to `sqrt(a^2 + b^2) - a`