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Question - 8

$</p><p></p><p>\rightarrow \quad \frac{ 2^{30} + 2^{29} + 2^{28} }{ 2^{31} + 2^{30} + 2^{29} } = \frac{1}{2}</p><p></p><p>\\ \\</p><p></p><p>\mathbb{ Solving \: L.H.S} </p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad \frac{ 2^{28} \cancel{ \left( 2^{2} + 2 + 1 \right)} }{ 2^{29} \cancel{ \left( 2^{2} + 2 + 1 \right) }}</p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad \frac{2^{28}}{ 2^{29} } </p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad 2^{28 - 29} \qquad \quad \left( \because \frac{ a^{m} }{ a ^{n}} = a^{m - n} \right)</p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad 2^{-1} </p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad \frac{1}{2} \qquad \quad \left( \because a^{-m} = \frac{1}{a^{m}} \right) </p><p></p><p>\\ \\</p><p></p><p>\boxed{ \bold{ L.H.S = R.H.S } } </p><p></p><p>\\ \\</p><p></p><p>\bold{Hence, \: Proved} </p><p></p><p>$