Question

Using suitable identity, evaluate $(42{)}^3-(18{)}^3-(24{)}^3$

• A. $-54432$
• B. $54432$
• C. $-6880$
• D. $6880$

Answer 1

Answer: (B)

$54432$

The given expression $(42{)}^3-(18{)}^3-(24{)}^3$ can be rewritten as $(42{)}^3+(-18{)}^3+(-24{)}^3$.

We know that one of the identity is $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$

We first find $x+y+z$ by substituting $x=42,y=-18,z=-24$ as follows:

$x+y+z=42-18-24=0$

Therefore, the above identity becomes:

$x^3+y^3+z^3-3xyz=0\times (x^2+y^2+z^2-xy-yz-zx)\Rightarrow x^3+y^3+z^3-3xyz=0\Rightarrow (42{)}^3+(-18{)}^3+(-24{)}^3-3(42\times -18\times -24)=0\Rightarrow (42{)}^3+(-18{)}^3+(-24{)}^3=3\times 18144\Rightarrow (42{)}^3-(18{)}^3-(24{)}^3=54432$

Hence, $(42{)}^3-(18{)}^3-(24{)}^3=54432$.