Question

$\frac{ {b}^{2} }{ \sqrt{ {a}^{2} + {b}^{2} } + a }$
Rationalize the above

Answer 1

Multiply numerator and denominator by
$\sqrt{ {a}^{2} + {b}^{2} } - a$
$\frac{ {b}^{2} \times ( \sqrt{ {a}^{2} + {b}^{2} } - a) }{ ( \sqrt{ {a}^{2} + {b}^{2} } + a) ( \sqrt{ {a}^{2} + {b}^{2} } - a)} \\ = \frac{ {b}^{2} \times ( \sqrt{ {a}^{2} + {b}^{2} } - a) }{( {a}^{2} + {b}^{2} ) - {a}^{2} } \\ = \frac{ {b}^{2} \times ( \sqrt{ {a}^{2} + {b}^{2} } - a)}{ {b}^{2} } \\ = \sqrt{ {a}^{2} + {b}^{2} } - a$