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Prove that
\( \left(a^{-1}+b^{-1}\right)^{-1}=\frac{a b}{a+b} \)
Given:
\( \left(a^{-1}+b^{-1}\right)^{-1}=\frac{a b}{a+b} \)
To do:
We have to prove that \( \left(a^{-1}+b^{-1}\right)^{-1}=\frac{a b}{a+b} \).
Solution:
We know that,
$(a^{m})^{n}=a^{m n}$
$a^{m} \times a^{n}=a^{m+n}$
$a^{m} \div a^{n}=a^{m-n}$
$a^{0}=1$
LHS $=(a^{-1}+b^{-1})^{-1}$
$=(\frac{1}{a}+\frac{1}{b})^{-1}$
$=(\frac{b+a}{a b})^{-1}$
$=\frac{a b}{a+b}$
$=$ RHS
Hence proved.
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