Question

Evaluate the following using the suitable identity.
(i) ${99}^3$
(ii)${1001}^2$
(iii) ${105}^3$
(iv) $105\times 108$
(v) $91\times 106$
(vi) ${107}^3$

Answer 1

$\left(1\right)(99{)}^3=(100-1{)}^3=(100{)}^3-3(100\left)\right(1\left)\right(100-1)-(1{)}^3$

$=1000000-29700-1$

$=970299$

$\left(2\right)(1001{)}^2=(1000+1{)}^2=(1000{)}^2+2(1000\left)\right(1)+(1{)}^2$

$=1000000+2000+1$

$=1002001$

$\left(3\right)(105{)}^3=(100+5{)}^3=(100{)}^3+100(5\left)\right(100+5)+5^3$

$=1000000+157500+125$

$=1157625$

$\left(4\right)105\times 108$

$=(100+5)(100+8)$

$=10000+800+500+40$

$11840$

$\left(5\right)91\times 106$

$=(100-9)(100+8)$

$=10000+600-900-54$

$=9646$

$\left(6\right)1073=(100+7{)}^3=(100{)}^3+7\left(100\right)\left(3\right)(100+7)+7^3$

$=1000000+224700+343$

$=1225043$