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Algebra Level 4

\[\begin{array}{} & \ \ \ (a+b+c+d)^4+(a+b-c-d)^4 \\ & +(a-b+c-d)^4+(a-b-c+d)^4 \\ & -(a+b+c-d)^4-(a+b-c+d)^4 \\ & -(a-b+c+d)^4-(-a+b+c+d)^4 \end{array}\] Simplify the above expression.

a b c d abcd a − b + c + d a-b+c+d 178 a b c d 178abcd 0 0 ( a b c d ) 4 (abcd)^4 192 a b c d 192abcd

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Rohit Udaiwal by Rohit Udaiwal , 20, India

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3 solutions

I assumed a = b = c = d a=b=c=d and then solved it.
( 4 a ) 4 + ( 0 a ) 4 + ( 0 a ) 4 + ( 0 a ) 4 − ( 2 a ) 4 − ( 2 a ) 4 − ( 2 a ) 4 − ( 2 a ) 4 (4a)^4+(0a)^4+(0a)^4+(0a)^4-(2a)^4-(2a)^4-(2a)^4-(2a)^4
( 4 a ) 4 − 4 [ ( 2 a ) 4 ] (4a)^4-4[(2a)^4]
256 a 4 − 4 [ 16 a 4 ] 256a^4-4[16a^4]
256 a 4 − 64 a 4 256a^4-64a^4
⇒ 192 a 4 \Rightarrow192a^4
Now, opening a 4 a^4 as a b c d abcd .
192 a b c d \boxed{192abcd}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

nice soln...

Atul Shivam - 5 years, 6 months ago

Congratulations for the smart approach.

Niranjan Khanderia - 5 years, 6 months ago

Ohhh!!! Very good method.

Natchayaporn Sindhurattavej - 5 years, 6 months ago

OR you could just observe that a b c d abcd is the only thing that survives .... but there are 8 of them that add up the only such multiple of 8 in options is 192 :) ... good solution though!

Abhinav Raichur - 5 years, 6 months ago

yeah..did same

Dev Sharma - 5 years, 6 months ago

But is this right process to solve this

Chandrasekhar Mantrabuddi - 5 years, 6 months ago

The answer has to be in abcd. It has to have some power of 2 0r 0 and a factor 3 for abcd. 178=89. can not have factor 89, nor could it be 1. Let w=a+b, x=a-b, y=c+d, z=c-d, we see it can not be 0. Only middle term remains BY INSPECTION. So 192abcd is the choice from the given list. The other two methods given are fare better. But just to show another approach I have given above.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Mohit Gupta
Dec 12, 2015

Well its a bashed soln. However it saves your time a lot....let a=1 b=2 c=-1 d=-2..... Now its merely a one min que... :)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Haha we chose the smart way ... Saves time...

A Former Brilliant Member - 5 years, 6 months ago

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