Question

The product of \[\left( {4x - 3y} \right)\] and \[\left( {16{x^2} + 12xy + 9{y^2}} \right)\] is
\[\left( 1 \right)\] \[{\left( {4x + 9y} \right)^3}\]
\[\left( 2 \right)\] \[{\left( {16{x^2} - 6xy + 9{y^2}} \right)^2}\]
\[\left( 3 \right)\] \[64{x^3} - 27{y^3}\]
\[\left( 4 \right)\] none of these

Answer 1

Hint: We have to find the value of the product of the given two expressions. We solve this question using the concept of the formula of the difference of the cubes of two numbers. First, we will simplify the given expression of the product in terms such that the product can be written for the terms of the formula of the difference of the cubes of two numbers. Then using the formula and simplifying the expression we can obtain the simplified value of the product of the two expressions.

Complete step-by-step solution:
Given:
Product of \[\left( {4x - 3y} \right)\] and \[\left( {16{x^2} + 12xy + 9{y^2}} \right)\]
Let us consider that \[P\] is the product of the two expressions.
Now, we can write the expression for the product as:
\[P = \left( {4x - 3y} \right)\left( {16{x^2} + 12xy + 9{y^2}} \right)\]
As we have two variables as \[4x\] and \[3y\] we will simplify the expression such that we obtain a relation for the formula of difference of cubes of two numbers.
The formula for the difference of cubes of two numbers is given as:
\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)\]
Now, on simplifying we can write the expression as:
\[P = \left( {4x - 3y} \right)\left( {{{\left( {4x} \right)}^2} + \left( {4x} \right)\left( {3y} \right) + {{\left( {3y} \right)}^2}} \right)\]
Thus, on comparing the two expressions the product of the expressions and the formula of the difference of cubes of two numbers, we can write the product as:
\[P = {\left( {4x - 3y} \right)^3}\]
Hence, the product of \[\left( {4x - 3y} \right)\] and \[\left( {16{x^2} + 12xy + 9{y^2}} \right)\] is \[{\left( {4x + 9y} \right)^3}\].
Thus, the correct option is \[\left( 1 \right)\].

Note: We can also solve this question by expanding the terms of the product by actual multiplication and then simplifying the terms and then further using the methods of solving the equations for obtain its roots so as to simply the expression, but it would be a time consuming and lengthy method. So, we prefer the method as done above for solving such types of questions.