Question

Simplify ${{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}$
$\begin{align}
  & A.\text{ 120}{{\text{x}}^{\text{2}}}{{\text{y}}^{\text{2}}}\text{+250}{{\text{y}}^{\text{3}}} \\
 & \text{B}\text{. -120}{{\text{x}}^{\text{2}}}\text{y+250}{{\text{y}}^{\text{3}}} \\
 & \text{C}\text{. 120}{{\text{x}}^{\text{2}}}\text{y+250}{{\text{y}}^{\text{3}}} \\
 & \text{D}\text{. -120}{{\text{x}}^{\text{2}}}\text{y-250}{{\text{y}}^{\text{3}}} \\
\end{align}$

Answer 1

Hint: In such a type of question simplify the given expression by expanding the given expression. This is also called involution. Involution is the general name for multiplying an expression by itself so as to find its second, third, fourth, or any other power. In this case we use the formula ${{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3{{a}^{2}}b$and the formula ${{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3{{a}^{2}}b$.Assume $a=2x$and $b=5y$. Substitute the value of $a$and $b$in the expression and then subtract as per the question. Here we have to cancel out the common terms and get the result.


Complete step-by-step answer:
First of all, we write the given expression which we have to simplify, we have to simplify
${{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}$
Let us assume that $a=2x$ and $b=5y$
Now we have the formula
${{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3{{a}^{2}}b$
Hence, we can write
\[{{(2x-5y)}^{3}}={{\left( 2x \right)}^{3}}-{{\left( 5y \right)}^{3}}-3{{(2x)}^{2}}(5y)+3(2x){{(5y)}^{2}}\]
Further we can write the above expression as
\[{{(2x-5y)}^{3}}=8{{x}^{3}}-125{{y}^{3}}-60{{x}^{2}}y+150x{{y}^{2}}---(a)\]
Similarly, we can expand the expression${{(2x+5y)}^{3}}$as
\[{{(2x+5y)}^{3}}={{\left( 2x \right)}^{3}}+{{\left( 5y \right)}^{3}}+3{{(2x)}^{2}}(5y)+3(2x){{(5y)}^{2}}\]
Further, we can write the above expression as
\[{{(2x+5y)}^{3}}=8{{x}^{3}}+125{{y}^{3}}+60{{x}^{2}}y+150x{{y}^{2}}---(b)\]
As from question we have to find out the difference ${{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}$
So, in order to find out we have to subtract expression$(b)$ from expression $(a)$
So, we can write
$\begin{align}
  & {{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}=8{{x}^{3}}-125{{y}^{3}}-60{{x}^{2}}y+150x{{y}^{2}}-(8{{x}^{3}}+125{{y}^{3}}+60{{x}^{2}}y+150x{{y}^{2}}) \\
 & \Rightarrow {{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}=8{{x}^{3}}-125{{y}^{3}}-60{{x}^{2}}y+150x{{y}^{2}}-8{{x}^{3}}-125{{y}^{3}}-60{{x}^{2}}y-150x{{y}^{2}} \\
\end{align}$
On cancelling the common terms, we can write further
$\Rightarrow {{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}=-125{{y}^{3}}-60{{x}^{2}}y-125{{y}^{3}}-60{{x}^{2}}y$
So further we can write
$\Rightarrow {{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}=-250{{y}^{3}}-120{{x}^{2}}y$
This can be written as
$\Rightarrow {{(2x-5y)}^{3}}-{{(2x+5y)}^{3}}=-120{{x}^{2}}y-250{{y}^{3}}$
Hence, we see here option D is correct.


Note: This question can also be solved as by assuming that
$a=2x-5y $ and $b=2x+5y$, then using the formula ${{a}^{3}}-{{b}^{3}}=(a-b)({{a}^{2}}+ab+{{b}^{2}})$
Here we use the term expression and not equation. A math expression is different from a math equation. An equation will always use an equivalent operator. That is in equation we have two side
Left- hand side and right-hand side with the sign of equality (=). The expression is a finite combination of symbols that is well formed according to rules that depend on the context.
The expression has no equality sign.