Question

Simplify \[{(2x - 5y)^3} - {(2x + 5y)^3}\]?

Answer 1

Hint: Here in this there are algebraic identities, we use the formulas of algebraic identities and on simplifying the terms we obtain the required solution for the given question. Here we use the two algebraic identities and they are \[{(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)\] and \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]

Complete step-by-step solution:
The algebraic expression is a combination of the constants, variables and it includes the arithmetic operation symbols. In algebraic expression we have three different kinds namely, monomial, binomial and polynomial expressions. We have some algebraic identities for some algebraic expression

Now consider the given expression which is present in the question
\[{(2x - 5y)^3} - {(2x + 5y)^3}\]
Here in this question we are applying the subtraction operation on the two algebraic identities
Expand these by using the standard algebraic identities and that is given by
\[{(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)\] and \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]
On using these identities and given expression is written as
\[ \Rightarrow {(2x)^3} - {(5y)^3} - 3(2x)(5y)(2x - 5y) - ({(2x)^3} + {(5y)^3} + 3(2x)(5y)(2x + 5y))\]
On simplifying each term, we write the given expression as
\[ \Rightarrow 8{x^3} - 125{y^3} - 30xy(2x - 5y) - (8{x^3} + 125{y^3} + 30xy(2x + 5y))\]
For the second term we should multiply the minus sign and so we have
\[ \Rightarrow 8{x^3} - 125{y^3} - 30xy(2x - 5y) - 8{x^3} - 125{y^3} - 30xy(2x + 5y))\]
Here some terms will gets cancels and so we have
\[ \Rightarrow - 125{y^3} - 30xy(2x - 5y) - 125{y^3} - 30xy(2x + 5y))\]
On further simplifying we have
\[ \Rightarrow - 125{y^3} - 60{x^2}y + 150x{y^2} - 125{y^3} - 60{x^2}y - 150x{y^2}\]
cancelling the terms which are get to cancel, and we have
\[ \Rightarrow - 125{y^3} - 60{x^2}y - 125{y^3} - 60{x^2}y\]
On adding the like terms we have
\[ \Rightarrow - 250{y^3} - 120{x^2}y\]
This cannot be further simplified, because they both are unlike terms
now we can take -10y as a common and we can rewrite the above expression and we have
\[ \Rightarrow - 10y(25{y^2} + 12{x^2})\]
Hence this is the simplified form of the given expression.

Thus the final expression is \[ - 10y(25{y^2} + 12{x^2})\].

Note: Since the question contains the algebraic expression and formula of the algebraic identities, it is easy to solve the problem if we knew the standard algebraic formulas. We should take care of signs because it may sometimes be while solving. The like terms can be cancelled or added but not unlike terms.