Question

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

Answer 1

(i)$(99{)}^3$

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

We know that$(a-b{)}^3=a^3-b^3-3ab(a-b)$ (1)

In this problem, we have a=100 and b=1, putting these values in (1), we get

$(100-1{)}^3=(100{)}^3-(1{)}^3-3(100\left)\right(1\left)\right(100-1)=1000000-1-29700=970299$

(ii)$(102{)}^3$

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

We know that $(a+b{)}^3=a^3+b^3+3ab(a+b)$ (1)

In this problem, we have a=100 and b=2, putting these values in (1), we get

$(100+2{)}^3=(100{)}^3+(2{)}^3+3(100\left)\right(2\left)\right(100+2)=1000000+8+61200=1061208$

(iii)$(998{)}^3$

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

We know that $(a-b{)}^3=a^3-b^3-3ab(a-b)$ (1)

In this problem, we have a=1000 and b=2, putting these values in (1), we get

$(1000-2{)}^3=(1000{)}^3-(2{)}^3-3(1000\left)\right(2\left)\right(1000-2)=1000000000-8-5988000=994011992$