Evaluate (using factors): $ {301^2} \times 300 - {300^3} $

Hint: First of find the common multiple and take common from both the terms and then use identities of the quadratic equations $ {a^2} - {b^2} = (a - b)(a + b) $ and then simplify for the resultant value.

Complete step-by-step answer:
Take the given expression –
 $ {301^2} \times 300 - {300^3} $
By using the additive law for power and exponents which states that when bases are equal then powers are added when there is multiplicative sign in between the two terms. Such as \[{x^{ab}} = {x^a} \times {x^b}\]
We can rewrite the above expression as –
 $ = {301^2} \times 300 - {300^2} \times 300 $
We can observe that $ 300 $ is the common multiple in both the terms, so take it common
 $ = 300[{301^2} - {300^2}] $
The above expression is in the form of a difference of two squares. So, we can use the identity $ {a^2} - {b^2} = (a - b)(a + b) $
 $ = 300[(301 - 300)(301 + 300)] $
Simplify the above equation –
 $
   = 300[(1)(601)] \\
   = 300(601) \;
  $
Do multiplication –
 $ {301^2} \times 300 - {300^3} = 180,300 $
This is the required solution.
So, the correct answer is “180,300”.

Note: Remember different identities to solve the given quadratic equations for the efficient and the accurate solutions. Know the concepts of squares and square-root. Square is the number multiplied itself and cube it the number multiplied thrice.