Simplifying Expressions

Algebra Level 2

343 × 343 × 343 − 113 × 113 × 113 343 × 343 + 343 × 113 + 113 × 113 = ? \large \dfrac{343 \times343 \times343 - 113 \times113 \times113}{343\times343 + 343 \times113 + 113 \times113} = \, ?


The answer is 230.

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Anuj Shikarkhane by Anuj Shikarkhane , 19, India

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2 solutions

Rishabh Jain
Jun 30, 2016

Relevant wiki: Algebraic Identities

Substitute x = 343 , y = 113 \color{#D61F06}{x=343}, \color{#3D99F6}{y=113} followed by using a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) \color{teal}{a^3-b^3=(a-b)(a^2+ab+b^2)}

A = 343 × 343 × 343 − 113 × 113 × 113 343 × 343 + 113 × 343 + 113 × 113 \mathcal A=\dfrac{\color{#D61F06}{343}\times\color{#D61F06}{343} \times\color{#D61F06}{343} - \color{#3D99F6}{113} \times \color{#3D99F6}{113} \times \color{#3D99F6}{113} }{\color{#D61F06}{343}\times\color{#D61F06}{343} +\color{#3D99F6}{113}\times\color{#D61F06}{343} + \color{#3D99F6}{113} \times\color{#3D99F6}{113} }

= x 3 − y 3 x 2 + y x + y 2 =\dfrac{\color{#D61F06}{x}^3-\color{#3D99F6}{y}^3}{\color{#D61F06}{x}^2+\color{#3D99F6}{y}\color{#D61F06}{x}+\color{#3D99F6}{y}^2}

= ( x − y ) ( x 2 + y x + y 2 ) x 2 + y x + y 2 =\dfrac{(\color{#D61F06}{x}-\color{#3D99F6}{y})\cancel{(\color{#D61F06}{x}^2+\color{#3D99F6}{y}\color{#D61F06}{x}+\color{#3D99F6}{y}^2)}}{\cancel{\color{#D61F06}{x}^2+\color{#3D99F6}{y}\color{#D61F06}{x}+\color{#3D99F6}{y}^2}}

= x − y = 343 − 113 = 230 =\color{#D61F06}{x}-\color{#3D99F6}{y}=\color{#D61F06}{343}-\color{#3D99F6}{113}\large =\color{#69047E}{\boxed{230}}

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same solution.. +1 :)

Novril Razenda - 4 years, 11 months ago

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That's great..... :-)

Rishabh Jain - 4 years, 11 months ago

Nice use of colors, however, note that on the second line and third line y 2 \color{#3D99F6}{y}^2 is written as y 2 \color{#D61F06}{y}^2

Hung Woei Neoh - 4 years, 11 months ago

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Thanks.. I knew I forgot something in one of my solutions here it is... But it's very difficult to keep track of so many solutions... :-)

Rishabh Jain - 4 years, 11 months ago

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I know. These few days, I have been seeing the same two people dump out so many solutions

Hung Woei Neoh - 4 years, 11 months ago

Nice solution.. +1 :)

Sabhrant Sachan - 4 years, 11 months ago

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Thanks.... :-)

Rishabh Jain - 4 years, 11 months ago
Mohammad Khaza
Jul 2, 2017

suppose, 343=a and 113=b

so,it stands as..=a^3-b^3/(a^2 + ab + b^2) 

                 =(a-b)(a^2+ ab +b^2) /(a^2 + ab + b^2)

                   =a-b

                   =343-113=230
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