Cubes?

Algebra Level 2

201 5 3 200 1 3 1 4 3 2015 × 2001 × 14 = ? \large \dfrac{2015^{3} -2001^{3} - 14^{3}}{2015 \times2001 \times 14} = \ ?


The answer is 3.

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6 solutions

Rimson Junio
Aug 28, 2015

( a + b ) 3 a 3 b 3 ( a + b ) ( a ) ( b ) = 3 a 2 b + 3 a b 2 ( a + b ) ( a ) ( b ) = 3 a b ( a + b ) a b ( a + b ) = 3 \frac{(a+b)^3-a^3-b^3}{(a+b)(a)(b)}=\frac{3a^2b+3ab^2}{(a+b)(a)(b)}=\frac{3ab(a+b)}{ab(a+b)}=3 a = 2000 , b = 11 a=2000, b=11

Nice solution!!

Rmflute Shrivastav - 5 years, 9 months ago

Let a = 2011 , b = 2000 a=2011, b=2000 : ( a + b ) 3 a 3 b 3 ( a + b ) a b = a 3 + 3 a 2 b + 3 a b 2 + b 3 a 3 b 3 ( a + b ) a b \frac{(a+b)^3-a^3-b^3}{(a+b)ab}=\frac{a^3+3a^2b+3ab^2+b^3-a^3-b^3}{(a+b)ab}

= 3 a b ( a + b ) ( a + b ) a b = 3 = \frac{3\color{#D61F06}{ab(a+b)}}{\color{#D61F06}{(a+b)ab}}=\boxed{3}

Nice explanation I understood it well!

Rmflute Shrivastav - 5 years, 9 months ago
Sadasiva Panicker
Sep 10, 2015

If a+b+c = 0 Then a^3 + b^3 + c^3 = 3abc, So (2015^3 - 2001^3 - 14^3)/20015 x 2001 x 14 = 3x20015x-2001x-14//2001x2001x14 = 3

Let a a , b b and c c be real numbers such that a + b + c = 0 a+b+c=0 . This means a + b = c a 3 + 3 a b ( a + b ) + b 3 = c 3 a 3 + b 3 + c 3 = 3 a b c a+b=-c \rightarrow a^{3}+3ab(a+b)+b^{3}=-c^{3} \rightarrow a^{3}+b^{3}+c^{3}=3abc . So 2011 2000 11 = 0 201 1 3 200 0 3 1 1 3 2011 2000 11 = 3 2011 2000 11 2011 2000 11 = 3 2011-2000-11=0 \rightarrow \dfrac{2011^{3}-2000^{3}-11^{3}}{2011\cdot2000\cdot11} = \frac{3\cdot2011\cdot2000\cdot11}{2011\cdot2000\cdot11}=3 . And we have a property and the answer

Rajat Sella
Sep 10, 2015

a=2015 , b=2001 , c=14 since a-b = c ; (a-b)^3 = c^3 ; a^3 - b^3 - 3ab(a-b) = c^3 ; a^3 - b^3 - c^3 = 3ab(a-b) ; since a - b = c ; a^3 - b^3 - c^3 = 3abc ; putting a,b and c value ; (2015^3 - 2001^3 - 14^3 ) / 2015.2001.14 = 3 ; 3 is the answer

Sakshi Rathore
Aug 31, 2015

when a + b + c = 0 a+b+c=0 where a = 2011 , b = 2000 , c = 11 a=2011,b=-2000,c=-11 then a 3 + b 3 + c 3 = 3 a b c a^3+b^3+c^3=3abc here abc is in denominator so simply the answer becomes 3 3

Better.....

subha chakraborty - 5 years, 9 months ago

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