Evaluate the following using suitable identities: <br/><br/>(i) $(99)^{3}$ <br/><br/>(ii) $(102)^{3}$ <br/><br/>(iii) $(998)^{3}$

Solution:

It is known that,

$(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)$

and $(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)$

(i) $(99)^{3}=(100-1)^{3}$

$=(100)^{3}-(1)^{3}-3(100)(1)(100-1)$

$=1000000-1-300(99)$

$=1000000-1-29700$

$=970299$

(ii) $(102)^{3}=(100+2)^{3}$

$=(100)^{3}+(2)^{3}+3(100)(2)(100+2)$

$=1000000+8+600(102)$

$=1000000+8+61200$

$=1061208$

(iii) $(998)^{3}=(1000-2)^{3}$

$=(1000)^{3}-(2)^{3}-3(1000)(2)(1000-2)$

$=1000000000-8-6000(998)$

$=1000000000-8-5988000$

$=1000000000-5988008$

$=994011992$

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