SSC CGL 24 Aug 2021 Shift 2 Paper

Question : 62 of 100

Marks: +1, -0

62. Simplify the following expression.

\;{5(a^{6}-b^{6})^{3}+5(b^{6}-c^{6})^{3}+5(c^{6}-a^{6})^{3}}/{2(a^{3}-b^{3})^{3}+2(b^{3}-c^{3})^{3}+2(c^{3}-a^{3})^{3}}

Solution:

\;{5(a^{6}-b^{6})^{3}+5(b^{6}-c^{6})^{3}+5(c^{6}-a^{6})^{3}}/{2(a^{3}-b^{3})^{3}+2(b^{3}-c^{3})^{3}+2(c^{3}-a^{3})^{3}}

We know that, if

a+b+c=0

then

a^{3}+b^{3}+c^{3}=3 a b c

So, here

a^{6}-b^{6}+b^{6}-c^{6}+c^{6}-a^{6}=0

Put

3(a^{6}-b^{6})(b^{6}-c^{6})(c^{6}-a^{6})

in place of numerator similar in denominator.

=\;{5}/{2}[\;{3(a^{6}-b^{6})(b^{6}-c^{6})(c^{6}-a^{6})}/{3(a^{3}-b^{3})(b^{3}-c^{3})(c^{3}-a^{3})}]

=\;{5}/{2}[\table (a^{3}-b^{3})(a^{3}+b^{3}) ×(b^{3}+c^{3}) ; \;{(b^{3}-c^{3}) ×(c^{3}+a^{3})(c^{3}-a^{3})}/{(a^{3}-b^{3})(b^{3}-c^{3})(c^{3}-a^{3})} ; {[ ∵ (a+b)(a-b)=a^{2}-b^{2}]}].

=\;{5}/{2}(a^{3}+b^{3})(b^{3}+c^{3})(c^{3}+a^{3})

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