Another unconscious algebra!

Algebra Level 3

1 a 2 ( a 6 + 3 a 4 b − 2 + a 2 b 4 3 − 1 + b 6 ) 2 / 3 + [ ( b 2 / 3 − a 2 / 3 ) 3 − 2 a 2 − b 2 a 2 + ( b 2 / 3 − a 2 / 3 ) 3 + 2 b 2 ] − 3 \frac { 1 }{ { a }^{ 2 } } \sqrt { { \left( { a }^{ 6 } + \frac { 3{ a }^{ 4 } }{ { b }^{ -2 } } + \frac{ { a }^{ 2 }{ b }^{ 4 } }{ { 3 }^{ -1 } } + { b }^{ 6 } \right) }^{ 2/3 } } + { \left[ \frac { { \left( { b }^{ 2/3 }-{ a }^{ 2/3 } \right) }^{ 3 }-2{ a }^{ 2 }-{ b }^{ 2 } }{ { a }^{ 2 } + { \left( { b }^{ 2/3 }-{ a }^{ 2/3 } \right) }^{ 3 }+2{ b }^{ 2 } } \right] }^{ -3 }

Find the value of expression above for a = 2016 a = 2016 and b = 2017 b = 2017 .


The answer is 1.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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1 solution

X = 1 a 2 ( a 6 + 3 a 4 b − 2 + a 2 b 4 3 − 1 + b 6 ) 2 3 + ( ( b 2 3 − a 2 3 ) 3 − 2 a 2 − b 2 a 2 + ( b 2 3 − a 2 3 ) 3 + 2 b 2 ) − 3 = ( a 6 + 3 a 4 b 2 + 3 a 2 b 4 + b 6 ) 1 3 a 2 + ( a 2 + ( b 2 3 − a 2 3 ) 3 + 2 b 2 ( b 2 3 − a 2 3 ) 3 − 2 a 2 − b 2 ) 3 = ( a 2 + b 2 ) 3 × 1 3 a 2 + ( a 2 + b 2 − 3 b 4 3 a 2 3 + 3 b 2 3 a 4 3 − a 2 + 2 b 2 b 2 − 3 b 4 3 a 2 3 + 3 b 2 3 a 4 3 − a 2 − 2 a 2 − b 2 ) 3 = a 2 + b 2 a 2 + ( 3 b 2 − 3 b 4 3 a 2 3 + 3 b 2 3 a 4 3 − 3 b 4 3 a 2 3 + 3 b 2 3 a 4 3 − 3 a 2 ) 3 = a 2 + b 2 a 2 + ( b 2 − b 4 3 a 2 3 + b 2 3 a 4 3 − a 2 − b 4 3 a 2 3 + b 2 3 a 4 3 ) 3 = a 2 + b 2 a 2 + b 2 a 2 ( b 4 3 − b 2 3 a 2 3 + a 4 3 − a 4 3 − b 4 3 + b 2 3 a 2 3 ) 3 = a 2 + b 2 a 2 − b 2 a 2 = a 2 a 2 = 1 \begin{aligned} X & = \frac 1{a^2} \sqrt{\left(a^6+\frac {3a^4}{b^{-2}}+\frac {a^2b^4}{3^{-1}} + b^6 \right)^\frac 23} + \left(\frac {\left(b^\frac 23 -a^\frac 23 \right)^3 - 2a^2-b^2}{a^2+\left(b^\frac 23 -a^\frac 23 \right)^3+2b^2} \right)^{-3} \\ & = \frac {(a^6+3a^4b^2+3a^2b^4+b^6) ^\frac 13}{a^2} + \left(\frac {a^2+\left(b^\frac 23 -a^\frac 23 \right)^3+2b^2}{\left(b^\frac 23 -a^\frac 23 \right)^3 - 2a^2-b^2} \right)^3 \\ & = \frac {(a^2+b^2)^{3 \times \frac 13}}{a^2} + \left(\frac {a^2+ b^2 - 3b^\frac 43 a^\frac 23 + 3b^\frac 23 a^\frac 43 - a^2 +2b^2}{b^2 - 3b^\frac 43 a^\frac 23 + 3b^\frac 23 a^\frac 43 - a^2 - 2a^2-b^2} \right)^3 \\ & = \frac {a^2+b^2}{a^2} + \left(\frac {3b^2 - 3b^\frac 43 a^\frac 23 + 3b^\frac 23 a^\frac 43}{- 3b^\frac 43 a^\frac 23 + 3b^\frac 23 a^\frac 43 - 3a^2} \right)^3 \\ & = \frac {a^2+b^2}{a^2} + \left(\frac {b^2 - b^\frac 43 a^\frac 23 + b^\frac 23 a^\frac 43}{-a^2 - b^\frac 43 a^\frac 23 + b^\frac 23 a^\frac 43} \right)^3 \\ & = \frac {a^2+b^2}{a^2} + \frac {b^2}{a^2} \left(\frac {b^\frac 43-b^\frac 23 a^\frac 23 + a^\frac 43}{-a^\frac 43-b^\frac 43 + b^\frac 23 a^\frac 23} \right)^3 \\ & = \frac {a^2+b^2}{a^2} - \frac {b^2}{a^2} \\ & = \frac {a^2}{a^2} = \boxed{1} \end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

@Chew-Seong Cheong

Nicely done.

At the first sight, do you thought that its going to be tough simplification?

Priyanshu Mishra - 4 years, 6 months ago

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Yes, I solved it by substituting in the numbers then only solved it by factoring.

Chew-Seong Cheong - 4 years, 6 months ago

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Help me in this also

https://brilliant.org/problems/telescopic-generations/?ref_id=1293707/

Priyanshu Mishra - 4 years, 6 months ago

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@Priyanshu Mishra – Your solution is wrong. It should be 1948 + 1949 = 3897 1948+1949=\boxed{3897} . Please change it so that I can enter a solution.

Chew-Seong Cheong - 4 years, 6 months ago

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@Chew-Seong Cheong – But its answer is 2432 only. Its not 3897

Priyanshu Mishra - 4 years, 6 months ago

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@Priyanshu Mishra – Sorry, you are right. The answer is 1948 2 1949 = 487 2 1945 \dfrac {1948}{\sqrt{2^{1949}}} = \dfrac {487}{\sqrt{2^{1945}}} . I forgot to simplify it.

Chew-Seong Cheong - 4 years, 6 months ago

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@Chew-Seong Cheong – So, are you now posting the solution?

Priyanshu Mishra - 4 years, 6 months ago

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@Priyanshu Mishra – Yes, doing it now.

Chew-Seong Cheong - 4 years, 6 months ago

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@Chew-Seong Cheong – Please help me to align my solution to extreme left of this problem:

Easier Trigonometry

Priyanshu Mishra - 4 years, 6 months ago

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@Priyanshu Mishra – [link title] (link address), with no space between ] and (. For example: Easier Trigonometry .

Chew-Seong Cheong - 4 years, 6 months ago

@Chew-Seong Cheong,

help me in this one also link

Priyanshu Mishra - 4 years, 6 months ago

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