Question

evolute (997)3 using identity​

Answer 1

Here are some identities useful in solving Question like above (one) :

$\bullet \: \rm(x + y)(x + y) = {x}^{2} + 2xy + {y}^{2}\\ \bullet \: \rm (x - y)(x - y) = {x}^{2} - 2xy + {y}^{2} \\ \small\bullet \: \rm (x + a)(x + b) = {(x)}^{2} + (a + b)x + ab \\ \bullet \: \rm (x - y)(x + y) = {(x)}^{2} - {(y)}^{2} \\ \small\bullet \: \rm(x + y)(x + y)(x + y) = {x}^{3} + {y}^{3} + {3x}^{2} y + {3xy}^{2} \\ \small\bullet \: \rm(x - y)(x - y)(x - y) = {x}^{3} - {y}^{3} - 3xy(x - y)$

Now,

▪Simplifying the (given) expression as :

$\tt \huge \mapsto \: \: \: {(997)}^{3} \\ \ \\ \huge \mapsto \: \: \sf( {1000 - 3)}^{3} \\ \\ \small \to \: \sf {(1000)}^{3} - {(3)}^{3} - 3(1000)(3) \{ 1000 - 3\} \\ \to \: \sf 1000000000 - 27 - 9000 \{ 1000 - 3\} \\ \to \: \sf 1000000000 - 27 - 9000000 + 27000 \\ \small \to \: \sf1000027000 - 9000027 \\ \to \: \sf \red{ 991026973}$

Hence,

• $\tt {(997)}^{3} = \red{991026973}$