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The value of ${{\text{a}}^4} - {{\text{b}}^4}$ is

  (a){\text{ (}}{{\text{a}}^2}{\text{ - }}{{\text{b}}^2}{\text{)(a + b)(a - b)}} \\
  {\text{(b) (}}{{\text{a}}^2}{\text{ - }}{{\text{b}}^2}{\text{)(a - b)(a - b)}} \\
  (c){\text{ (}}{{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{)(a + b)(a - b)}} \\
  ({\text{d}}){\text{ (}}{{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{)(a + b}}{{\text{)}}^2} \\

Last updated date: 16th Jul 2024
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HINT- In order to solve such types of questions we should keep one thing in mind that if both terms are perfect squares then use the difference of square formula.

Complete step-by-step answer:
In algebra, there is a formula known as the Difference of two squares:$({{\text{m}}^2}{\text{ - }}{{\text{n}}^2}) = ({\text{m + n}})({\text{m - n}})$ --(1)
Since both terms are perfect squares in the given question, factor using the difference of squares formula
In the case of ${{\text{a}}^4} - {{\text{b}}^4}$, you will see that ${{\text{a}}^4}$ is just ${({{\text{a}}^2})^2}$ and ${{\text{b}}^4}$is just ${({{\text{b}}^2})^2}$
${\text{ = }}{{\text{a}}^4} - {{\text{b}}^4} = {({{\text{a}}^2}{\text{)}}^2} - {({{\text{b}}^2}{\text{)}}^2}$ ---- (2)
Here, ${\text{m = }}({{\text{a}}^2}{\text{) and n = }}({{\text{b}}^2}{\text{)}}$
So using expression (1)
$ = {({{\text{a}}^2}{\text{)}}^2} - {({{\text{b}}^2}{\text{)}}^2} = \left( {({{\text{a}}^2}{\text{) + }}({{\text{b}}^2}{\text{)}}} \right)\left( {({{\text{a}}^2}{\text{) - }}({{\text{b}}^2}{\text{)}}} \right)$ --- (3)
But as you can see, we can use the formula (1) again in 2nd term of RHS
where ${\text{m = }}({\text{a) and n = }}({\text{b)}}$
$ = {\text{ }}({{\text{a}}^2}{\text{ - }}{{\text{b}}^2}) = ({\text{a + b}})({\text{a - b}})$--- (4)
On putting value of (4) in expression (3)
$ = {\text{ }}{({{\text{a}}^2}{\text{)}}^2} - {({{\text{b}}^2}{\text{)}}^2} = \left( {({{\text{a}}^2}{\text{) + }}({{\text{b}}^2}{\text{)}}} \right)\left( {({\text{a + b}})({\text{a - b}})} \right)$ ---- (5)
On putting value of (2) in expression (5)
\[ = {\text{ }}({{\text{a}}^4}{\text{)}} - ({{\text{b}}^4}{\text{)}} = \left( {({{\text{a}}^2}{\text{) + }}({{\text{b}}^2}{\text{)}}} \right)\left( {({\text{a + b}})({\text{a - b}})} \right)\]
Which is the required answer.
Hence option c is correct.

Note- Whenever we face such types of problems the key concept we have to remember is that we should always try to factorise those binomial expressions which are having even power using identity which is stated above. Sometimes we have to apply identity more than one, like in the above question, we used the difference of square identity twice.