Question

# The value of ${{\text{a}}^4} - {{\text{b}}^4}$ is$(a){\text{ (}}{{\text{a}}^2}{\text{ - }}{{\text{b}}^2}{\text{)(a + b)(a - b)}} \\ {\text{(b) (}}{{\text{a}}^2}{\text{ - }}{{\text{b}}^2}{\text{)(a - b)(a - b)}} \\ (c){\text{ (}}{{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{)(a + b)(a - b)}} \\ ({\text{d}}){\text{ (}}{{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{)(a + b}}{{\text{)}}^2} \\$

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.

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)
$= {\text{ }}({{\text{a}}^4}{\text{)}} - ({{\text{b}}^4}{\text{)}} = \left( {({{\text{a}}^2}{\text{) + }}({{\text{b}}^2}{\text{)}}} \right)\left( {({\text{a + b}})({\text{a - b}})} \right)$