I see i i

Algebra Level 2

( i + i i i ) 2 = ? \Large \left( \frac{\sqrt{i}+i\sqrt{i}}{i}\right)^2 \!= \, ?

0 2 3 i 2 i -2i -1

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5 solutions

Chew-Seong Cheong
Jan 28, 2016

( i + i i i ) 2 = ( i + i i i i ) 2 i in the nominator and denominator cancel off. = ( 1 + i i ) 2 = 2 i i = 2 \begin{aligned} \left(\frac{\sqrt{i} + i\sqrt{i}}{i}\right)^2 & = \left(\frac{\color{#D61F06}{\sqrt{i}} + i\color{#D61F06}{\sqrt{i}}}{\color{#D61F06}{\sqrt{i}} \sqrt{i}} \right)^2 \quad \quad \small \color{#D61F06}{\sqrt{i} \text{ in the nominator and denominator cancel off.}} \\ & = \left(\frac{1 + i}{\sqrt{i}}\right)^2 \\ & = \frac{2i}{i} \\ & = \boxed{2} \end{aligned}

(i+1)whole square =2i HOW?

Manan Maheshwari - 5 years, 4 months ago

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I understood how. Sorry for asking such a silly question

Manan Maheshwari - 5 years, 4 months ago

I'm sorry, this might seem silly, but I didn't understand what you've done. Can you explain please (I know how to solve the problem, just don't understand the way you algebraically manipulated the terms).

A Former Brilliant Member - 5 years, 4 months ago

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I have added comments in the solution. Do you understand now?

Chew-Seong Cheong - 5 years, 4 months ago

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Thank you very much! Now I can understand.

A Former Brilliant Member - 5 years, 4 months ago

Indeed... their answer is wrong

Phenyo Alphius - 5 years, 4 months ago

Brilliant +1

Ahmed Obaiedallah - 5 years, 4 months ago

yeah its simple thanks :p

zubair ahmad - 5 years, 4 months ago

your answer is the most simple lol

Jason Chrysoprase - 5 years, 4 months ago
Kay Xspre
Jan 27, 2016

Rewrite it into the form of ( 1 i + i ) 2 = i + 1 i + 2 = i i + 2 = 2 (\frac{1}{\sqrt{i}}+\sqrt{i})^2 = i+\frac{1}{i}+2 = i-i+2 = 2

( i + i i i ) 2 = i + i 3 + 2 i ( i ) 2 i 2 = i i + 2 i 2 i 2 = 2 \Large \left( \frac{\sqrt{i}+i\sqrt{i}}{i}\right)^2 = \dfrac{i+i^3+2i(\sqrt{i})^2}{i^2} = \dfrac{i-i+2i^2}{i^2} =2

i = e i π 2 i = e^{\frac{i\pi}{2}} ; ( i + i i i ) 2 = ( e i π 4 + e 3 i π 4 e i π 2 ) 2 = ( e i π 2 + 1 e i π 4 ) 2 = 2 i i = 2 (\frac{\sqrt{i} + i\sqrt{i}}{i} )^2 = ( \frac{e^{\frac{i\pi}{4}} + e^{\frac{3i\pi}{4}}}{e^{\frac{i\pi}{2}}})^2 = (\frac{e^{\frac{i\pi}{2}} + 1}{e^{\frac{i\pi}{4}}})^2 = \frac{2i}{i} = 2 or ( e i π 4 + e 3 i π 4 e i π 2 ) 2 = ( e i π 2 ( e i π 4 + e 3 i π 4 ) e i π 2 e i π 2 ) 2 = ( e i π 4 + e i π 4 ) 2 = ( 2 cos π 4 ) 2 = 2 ( \frac{e^{\frac{i\pi}{4}} + e^{\frac{3i\pi}{4}}}{e^{\frac{i\pi}{2}}})^2 = ( \frac{e^{\frac{-i\pi}{2}} \cdot (e^{\frac{i\pi}{4}} + e^{\frac{3i\pi}{4}})}{e^{\frac{-i\pi}{2}} \cdot e^{\frac{i\pi}{2}}})^2 = (e^{\frac{i\pi}{4}} + e^{\frac{-i\pi}{4}})^2 = (2 \cos \frac{\pi}{4})^2 = 2

( i + i i i ) 2 \Rightarrow \left( \frac{\sqrt{i}+i\sqrt{i}}{i} \right)^2

( i + i 3 i ) 2 \left(\frac{\sqrt{i}+\sqrt{i^3}}{i} \right)^2

( ( i ) 2 + ( i 3 ) 2 + 2 × i 4 ( i ) 2 ) \left(\frac{(\sqrt{i})^2+(\sqrt{i^3})^2+2×\sqrt{i^4}}{(i)^2}\right)

( i i + 2 × ( i ) 2 1 ) \left(\frac{i-i+2×(i)^2}{-1}\right)

( 2 ) ( 1 ) = 2 \frac{-(2)}{-(1)}=\boxed{2}

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