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7.Evaluate the following using suitable identities.
i)\({99}^3\)
ii)\({102}^3\)
iii)\({998}^3\)
Answer :

i)\({99}^3 = (100 - 1)^3 = (100)^3 - (1)^3 - 3(100)(1)(100 - 1)\)
by using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= 1000000 - 1 - 300(99)\)
\(= 1000000 - 1 - 2700)\)
\(= 970299\)

ii)\({102}^3 = (100 + 2)^3 = (100)^3 + (2)^3 + 3(100)(2)(100 + 2)\)
by using identity \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\)
\(= 1000000 + 8 + 600(102)\)
\(= 1000000 + 8 + 61200)\)
\(= 1061208\)

iii)\({998}^3 = (1000 - 2)^3 = (1000)^3 - (2)^3 - 3(1000)(2)(1000 - 2)\)
by using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= 1000000000 - 8 - 6000(998)\)
\(= 1000000000 - 1 - 5988000)\)
\(= 994011992\)