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Algebra Level 2

1 2 ÷ 1 2 ÷ 1 2 ÷ 1 2 ÷ 1 2 = ? \cfrac1{2 \div \cfrac1{2 \div \cfrac1{2\div \cfrac1{2\div \cfrac12 }}}} = \, ?

2 − 3 2^{-3} 2 − 4 2^{-4} 2 − 5 2^{-5} 2 − 6 2^{-6}

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Chung Kevin by Chung Kevin , 45, Australia

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2 solutions

1 2 ÷ 1 2 ÷ 1 2 ÷ 1 2 ÷ 1 2 = 1 2 ÷ 1 2 ÷ 1 2 ÷ 1 4 = 1 2 ÷ 1 2 ÷ 1 8 = 1 2 ÷ 1 16 = 1 32 = 2 − 5 \begin{aligned} \cfrac{1}{2\div\cfrac{1}{2\div\cfrac{1}{2\div\cfrac{1}{\color{#D61F06}{2\div\cfrac{1}{2}}}}}}&=\cfrac{1}{2\div\cfrac{1}{2\div\cfrac{1}{\color{#3D99F6}{2\div\cfrac{1}{4}}}}}\\ =\cfrac{1}{2\div\cfrac{1}{\color{#69047E}{2\div\cfrac{1}{8}}}}&=\cfrac{1}{\color{#EC7300}{2\div\cfrac{1}{16}}}\\ =\dfrac{1}{32}&=\boxed{2^{-5}}\end{aligned}

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Moderator note:

It wasn't immediately obvious how these fractions would play out, so we should track them carefully.

Thank you for your solution! Hope you enjoyed it!

Chung Kevin - 5 years, 2 months ago

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Thanks.It was a nice problem

Abdur Rehman Zahid - 5 years, 2 months ago

Such kind of problems are needed to be solved from bottom

A basic thing which is , a ÷ b = a × 1 b . a÷b = a×\frac {1}{b} .

Applying it here

2 ÷ 1 2 = 2 × 2 = 4 2÷\frac {1}{2} = 2×2=4 next , 2 ÷ 1 4 = 2 × 4 = 8..... 2÷\frac {1} {4}=2×4=8..... Continuing this we arrive at a final expression I.e 1 32 \frac {1}{32} which is nothing but 2 − 5 \boxed{2^{-5}} .

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