Question

rationalize the denominator a^2÷square root of a^2+b^2 +b

Answer 1

SOLUTION

TO DETERMINE

Rationalize the denominator

$\displaystyle \sf{ \frac{ {a}^{2} }{ \sqrt{ {a}^{2} + {b}^{2} } + b} }$

EVALUATION

$\displaystyle \sf{ \frac{ {a}^{2} }{ \sqrt{ {a}^{2} + {b}^{2} } + b} }$

$= \displaystyle \sf{ \frac{ {a}^{2} { \sqrt{ ({a}^{2} + {b}^{2} } - b)}}{ (\sqrt{ {a}^{2} + {b}^{2} } + b){ ( \sqrt{ {a}^{2} + {b}^{2} } - b)}} }$

$= \displaystyle \sf{ \frac{{a}^{2} ({ \sqrt{ {a}^{2} + {b}^{2} } - b)}}{ {( \sqrt{ {a}^{2} + {b}^{2} } )}^{2} - {b}^{2} } }$

$= \displaystyle \sf{ \frac{{a}^{2} ({ \sqrt{ {a}^{2} + {b}^{2} } - b)}}{ {a}^{2} + {b}^{2} - {b}^{2} } }$

$= \displaystyle \sf{ \frac{{a}^{2} ({ \sqrt{ {a}^{2} + {b}^{2} } - b)}}{ {a}^{2} } }$

$= \displaystyle \sf{ \sqrt{ {a}^{2} + {b}^{2} } - b }$

FINAL ANSWER

$\displaystyle \sf{ \frac{ {a}^{2} }{ \sqrt{ {a}^{2} + {b}^{2} } + b} = \sqrt{ {a}^{2} + {b}^{2} } - b}$

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