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Algebra Level 5

Find the value of -

511 − 24 2048 + 381 − 4 37892 − 21120 + 5 262144 − 196608 + 61440 ⋯ 6 5 4 3 \large \sqrt[3]{511-24\sqrt[4]{2048+381-4\sqrt[5]{37892-21120+5\sqrt[6]{262144-196608+61440\cdots}}}}


The answer is 7.

Vatsalya Tandon by Vatsalya Tandon , 21, India

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2 solutions

Vatsalya Tandon
Feb 13, 2016

Let f ( x ) f(x) be x − 1 x-1

f ( x ) = ( x − 1 ) 3 3 f(x) = \sqrt[3]{(x-1)^{3}}

f ( x ) = x 3 − 1 − 3 x ( x − 1 ) 3 f(x) = \sqrt[3]{x^3 - 1 - 3x(x-1)}

f ( x ) = x 3 − 1 − 3 x ( x − 1 ) 4 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{(x-1)^4}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 4 x + 1 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 - 4x + 1}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 4 x + 4 − 3 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 - 4x +4 - 3}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 3 − 4 ( x − 1 ) ) 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 -3 -4(x-1))}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 3 − 4 ( x − 1 ) 5 5 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 -3 -4\sqrt[5]{(x-1)^5}}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 3 − 4 x 5 − 5 x 4 + 10 x 3 − 10 x 2 + 5 x − 1 5 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 -3 -4\sqrt[5]{x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1}}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 3 − 4 x 5 − 5 x 4 + 10 x 3 − 10 x 2 + 4 + 5 x − 5 5 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 -3 -4\sqrt[5]{x^5 - 5x^4 + 10x^3 - 10x^2 + 4+ 5x - 5}}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 3 − 4 x 5 − 5 x 4 + 10 x 3 − 10 x 2 + 4 + 5 ( x − 1 ) 6 6 5 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 -3 -4\sqrt[5]{x^5 - 5x^4 + 10x^3 - 10x^2 + 4+ 5\sqrt[6]{(x-1)^6}}}}

f ( x ) = x 3 − 1 − 3 x x 4 − 4 x 3 + 6 x 2 − 3 − 4 x 5 − 5 x 4 + 10 x 3 − 10 x 2 + 4 + 5 x 6 − 6 x 5 + 15 x 4 . . . 6 5 4 3 f(x) = \sqrt[3]{x^3 - 1 - 3x\sqrt[4]{x^4 - 4x^3 + 6x^2 -3 -4\sqrt[5]{x^5 - 5x^4 + 10x^3 - 10x^2 + 4+ 5\sqrt[6]{x^6 - 6x^5 + 15x^4 ... }}}}

Plugging in x = 8 x=8 , we get-

f ( 8 ) = 512 − 1 − 24 4096 − 2048 + 384 − 3 − 4 3276 8 5 − 20480 + 5120 − 640 + 4 + 5 262144 − 196608 + 61440... 6 5 4 3 f(8) = \sqrt[3]{512 - 1 - 24\sqrt[4]{4096 - 2048 + 384 -3 -4\sqrt[5]{32768^5 - 20480 + 5120 - 640 + 4+ 5\sqrt[6]{262144 - 196608 + 61440 ... }}}}

f ( 8 ) = 511 − 24 2048 + 381 − 4 37892 − 21120 + 5 262144 − 196608 + 61440... 6 5 4 3 f(8) = \sqrt[3]{511 - 24\sqrt[4]{2048 + 381 -4\sqrt[5]{37892 - 21120 + 5\sqrt[6]{262144 - 196608 + 61440 ... }}}}

⇒ f ( 8 ) = 8 − 1 \Rightarrow f(8)= 8-1

7 \boxed{7}

11 Helpful 0 Interesting 1 Brilliant 0 Confused

Could you please increase the font? The radicals cannot be read. Nevertheless, Nice solution to a good question. Upvoted!

Somyaneel Sinha - 5 years, 3 months ago

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Thank You so much! Support of people like you makes me post problems and there solutions :)

Vatsalya Tandon - 5 years, 3 months ago
Prakhar Dwivedi
Feb 21, 2016

Good question but there is a trick to bypass the algorithm. One can guess the answer easily as 7 is the only whole no. cube root possible after subtracting multiples of 24 from 511 ;)

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes. That was my approach. Answer< 51 1 1 3 . 8 3 = 512. S o a n s w e r = 7. 511^{\frac 1 3}. ~~8^3=512.~~So~answer=7.

Niranjan Khanderia - 2 years, 10 months ago

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