Express each one of the following with rational denominator: b^2 … - Vidyadip

Question

Express each one of the following with rational denominator: $\frac{\text{b}^2}{\sqrt{\text{a}^2+\text{b}^2}+\text{a}}$

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Answer

$\frac{\text{b}^2}{\sqrt{(\text{a}^2+\text{b}^2)}+\text{a}}$Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor $\sqrt{\text{a}^2+\text{b}^2}-\text{a}$
$=\frac{\text{b}^2\big(\sqrt{\text{a}^2+\text{b}^2}-\text{a}\big)}{\big(\sqrt{(\text{a}^2+\text{b}^2)}+\text{a}\big)\big(\sqrt{(\text{a}^2+\text{b}^2)}-\text{a}\big)}$
As we know, $(\text{a}-\text{b})^2=(\text{a}^2-2\times\text{a}\times\text{b}+\text{b}^2)$
$=\frac{\text{b}^2\big(\sqrt{\text{a}^2+\text{b}^2}-\text{a}\big)}{(\text{a}^2+\text{b}^2)-\text{a}^2}$
$=\frac{\text{b}^2\big(\sqrt{\text{a}^2+\text{b}^2}-\text{a}\big)}{\text{b}^2}$

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