Question

Find the cube of $ \left( {x - 2y} \right) $

Answer 1

Hint: Cube of the number can be found by cubing the expression, then expanding it using cubic formula of $ {\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} $ . Compare the variables of the given expressions with the standard identities to get the required.

Complete step-by-step answer:
The given expression is
 $ E = \left( {x - 2y} \right) \cdots \left( 1 \right) $
We have to calculate $ {E^3} = {\left( {x - 2y} \right)^3} $ .
On comparing with it can be concluded that
 $ a = x $ and $ b = - 2y $
Now Cubing both sides of equation (1), we get [Using the formula of: $ {\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} $ ]
 $
\Rightarrow {E^3} = {x^3} + 3{\left( x \right)^2}\left( { - 2y} \right) + 3\left( x \right){\left( { - 2y} \right)^2} + {\left( { - 2y} \right)^3} \\
  {E^3} = {x^3} - 6{x^2}y + 12x{y^2} - 8{y^3} \\
  $
The answer for the cube of $ \left( {x - 2y} \right) $ is $ {x^3} - 6{x^2}y + 12x{y^2} - 8{y^3} $ .

Note: Care should be taken while substituting the values of and with their respective signs in the formula of . The minus sign should be carefully taken into account during the substitution.
It can also be solved using $ {\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3} $ .
If we can compare the expression $ {\left( {x - 2y} \right)^3} $ with $ {\left( {a - b} \right)^3} $ . We can say that $ a = x $ and $ b = 2y $ .
On expanding the expression $ {\left( {x - 2y} \right)^3} $ we get,
 $
  {E^3} = {x^3} - 3{\left( x \right)^2}\left( {2y} \right) + 3x{\left( {2y} \right)^2} - {\left( {2y} \right)^3} \\
  {E^3} = {x^3} - 6{x^2}y + 12x{y^2} - 8{y^3} \\
  $