Question

Simplify $ {\left( {3x + 4y} \right)^3} - {\left( {3x - 4y} \right)^3} - 216{x^2}y $

Answer 1

Hint: Let us consider 3x as a and 4y as b and replace them in the given expression. Then $ {\left( {3x + 4y} \right)^3} $ will be $ {\left( {a + b} \right)^3} $ and $ {\left( {3x - 4y} \right)^3} $ will be $ {\left( {a - b} \right)^3} $ . Expand these terms using the below mentioned formulas and subtract one from another. After subtracting you will be left with terms in a and b. Now replace a with 3x and b with 4y to get the required value.
Formulas used:
 $ {\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3} $
 $ {\left( {a - b} \right)^3} = {a^3} - 3ab\left( {a - b} \right) - {b^3} $

Complete step-by-step answer:
We are given to simplify $ {\left( {3x + 4y} \right)^3} - {\left( {3x - 4y} \right)^3} - 216{x^2}y $
Let 3x be a and 4y be b.
On substituting 3x as a and 4y as b in the given expression, the expression becomes $ {\left( {a + b} \right)^3} - {\left( {a - b} \right)^3} - 216{x^2}y $
On expanding the above cubes using the known formulas, we get
 $ {a^3} + 3ab\left( {a + b} \right) + {b^3} - \left( {{a^3} - 3ab\left( {a - b} \right) - {b^3}} \right) - 216{x^2}y $
 $ \Rightarrow {a^3} + 3ab\left( {a + b} \right) + {b^3} - {a^3} + 3ab\left( {a - b} \right) + {b^3} - 216{x^2}y $
On cancelling out similar the terms with different signs, we get
 $ \Rightarrow 3ab\left( {a + b} \right) + 2{b^3} + 3ab\left( {a - b} \right) - 216{x^2}y $
On multiplying 3ab with the terms inside the bracket, we get
 $ \Rightarrow 3{a^2}b + 3a{b^2} + 2{b^3} + 3{a^2}b - 3a{b^2} - 216{x^2}y $
On cancelling out the similar terms with different signs again, we get
 $ \Rightarrow 3{a^2}b + 2{b^3} + 3{a^2}b - 216{x^2}y $
 $ \Rightarrow 6{a^2}b + 2{b^3} - 216{x^2}y $
Now on substituting a as 3x and b as 4y in the above expression, we get $ \Rightarrow 6{\left( {3x} \right)^2}\left( {4y} \right) + 2{\left( {4y} \right)^3} - 216{x^2}y $
 $ \Rightarrow 6\left( {9{x^2}} \right)\left( {4y} \right) + 2\left( {64{y^3}} \right) - 216{x^2}y $
 $ \Rightarrow 216{x^2}y + 128{y^3} - 216{x^2}y $
 On cancelling out the similar terms with different signs, we get
 $ \Rightarrow 128{y^3} $
Therefore, the value of $ {\left( {3x + 4y} \right)^3} - {\left( {3x - 4y} \right)^3} - 216{x^2}y $ is $ 128{y^3} $
So, the correct answer is “$ 128{y^3} $”.

Note: Instead of writing the formulas of $ {\left( {a + b} \right)^3} $ and $ {\left( {a - b} \right)^3} $ separately and then subtracting them we can directly use the general formula of $ {\left( {a + b} \right)^3} - {\left( {a - b} \right)^3} $ , which is $ 6{a^2}b + 2{b^3} $ . Instead of considering 3x as a and 4y as b, we can solve the given problem by putting them as it is. We have done by replacing the terms as a and b to clear out confusion and get the result in a simple way.