Question

please answer my question
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Answer 1

$\text{\bf{\green{watch and learn How to Solve. }}}$

Question

$\sf \: \left( \dfrac{1}{1 + {a}^{n - m} }+ \dfrac{1}{1 + {a}^{m - n} } \right) \: \: is \: \: equal \: \: to$

Solution

${ \: \: \: \: \: \: : \: \implies\sf \: \dfrac{1}{1 + {a}^{n - m} }+ \dfrac{1}{1 + {a}^{m - n} }}$

• $\small\boxed{ \: \: \: \: \: \: \: \: \bf{ a^{x-y}=\frac{a^x}{a^y}} \: \: \: \: \: \: \: \: \: \: }$

${ \: \: \: \: \: \: : \: \implies\sf \: \dfrac{1}{1 + \frac{ {a}^{n} }{ {a}^{m} } }+ \dfrac{1}{1 + \frac{ {a}^{m} }{ {a}^{n} } }}$

• Taking LCM

${ \: \: \: \: \: \: : \: \implies\sf \: \dfrac{1}{ \frac{ {a}^{m } + {a}^{n} }{ {a}^{m} } }+ \dfrac{1}{ \frac{ {a}^{n} + {a}^{m} }{ {a}^{n} } }}$

${ \: \: \: \: \: \: : \: \implies\sf \: \dfrac{ {a}^{m} }{ {a}^{m} + {a}^{n} } + \dfrac{ {a}^{n} }{ {a}^{n} + {m}^{m} } }$

• Taking LCM of a^m + a^n

${ \: \: \: \: \: \: : \: \implies\sf \: \dfrac{ {a}^{m} + {a}^{n} }{ {a}^{m} + {a}^{n} } }$

${ \: \: \: \: \: \: : \: \implies\sf \: \cancel \dfrac{ {a}^{m} + {a}^{n} }{ {a}^{m} + {a}^{n} } }$

${ \: \: \: \: \: \: : \: \implies\sf \: 1 }$

$\gray{ \sf \: \left( \dfrac{1}{1 + {a}^{n - m} }+ \dfrac{1}{1 + {a}^{m - n} } \right) \: \: = \: \huge \bf1}$

• $\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \bigstar \: \underline{\bf{}}\\ {\boxed{\begin{array}{c | c} \frac{ \: ~~~~~~~~~~\: \: \: \: \:\sf Laws \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ } &\frac{ \: ~~~~~~~~~~ \: \: \: \: \:\sf Example \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ }\\ \sf \bigstar{a}^{m} \times {a}^{n} = {a}^{m + n} & \sf {a}^{2} \times {a}^{3} = {a}^{2 + 3} = {a}^{6} \\ \\ \sf \bigstar{a}^{m} \div {a}^{n} = {a}^{m - n}& \sf {a}^{3} \div {a}^{2} = {a}^{3 - 2} = {a}^{1} \\ \\ \sf{\bigstar \: \: \: \: \: \: ( {a}^{m} ) ^{n} = {a}^{mn} } & \sf( {a}^{2} ) ^{3} = {a}^{2 \times 3} = {a}^{6} \\ \\ {\bigstar\sf a {}^{m} \times {n}^{m} = (ab) ^{m} } &\sf a {}^{2} \times {b}^{2} = (ab) ^{2}\\ \\ \sf\bigstar \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {a}^{0} = 1& \sf {2}^{0} = 1 \: \: \: \: \\ \\ \sf \bigstar \: \: \: \: {\dfrac{ {a}^{m} }{ {b}^{m} }= \left( \dfrac{a}{b} \right) ^{m} }& \sf{\dfrac{ {a}^{2} }{ {b}^{2} }= \left( \dfrac{a}{b} \right) ^{2} }\\\\~~~~~~~~\bigstar\sf x^{\frac{m}{n} }=\sqrt[n]{x^m}\sf = (\sqrt[n]{x})^m & \sf x^{\frac{2}{3} }=\sqrt[3]{x^2} = (\sqrt[n]{x})^m\\ \\\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$