Question

`(a^2-b^2)/(a-b)-(a^3-b^3)/(a^2-b^2)=?`

i need how to start this question ty!

Answer 1

`(ab)/(a+b)`

Explanation:

`"simplify numerators/denominators using"`

`color(blue)"difference of squares"`

`•color(white)(x)a^2-b^2=(a-b)(a+b)`

`"and "color(blue)"difference of cubes"`

`•color(white)(x)a^3-b^3=(a-b)(a^2+ab+b^2)`

`rArr(cancel((a-b))(a+b))/cancel((a-b))-(cancel((a-b))(a^2+ab+b^2))/(cancel((a-b))(a+b))`

`=(a+b)-(a^2+ab+b^2)/(a+b)`

`"multiply and divide "(a+b)" by "(a+b)`

`=(a+b)^2/(a+b)-(a^2+ab+b^2)/(a+b)`

`"the fractions now have a "color(blue)"common denominator"`
`"so add/subtract numerators leaving the denominator"`

`=(cancel(a^2)+2abcancel(+b^2)cancel(-a^2)-abcancel(-b^2))/(a+b)`

`=(ab)/(a+b)`

Answer 2

`color(magenta)(=>(ab)/(a+b)`

Explanation:

`∙` `color(blue)[(a^2−b^2)]/(a−b)−color(red)[(a^3−b^3)]/ color(blue)[ (a^2−b^2)]`

`∙`Using the identities: `color(blue)(a^2-b^2=(a+b)(a-b)` and `color(red)(a^3-b^3= (a-b)(a^2+ab+b^2)`

`=> color(blue) [(a+b)(a-b)] /(a−b)-[color(red){(a-b)(a^2+ab+b^2)}]/color(blue)[(a+b)(a-b)]`

`=> [ (a+b) cancel ((a-b))] /cancel((a−b))-[cancel((a-b))(a^2+ab+b^2) ]/[(a+b)cancel((a-b))]`

`=> (a+b)-((a^2+ab+b^2))/((a+b))`

`∙`Multiplying the numerator and denominator of the first term with `(a+b)`, since the LCM is `(a+b)`:

`=>(a+b)^2/((a+b))-((a^2+ab+b^2))/((a+b))`

`∙` Using the identity:`color(darkorange)((a+b)^2=(a^2+2ab+b^2)`

`=>color(darkorange)((a^2+2ab+b^2))/((a+b))-((a^2+ab+b^2))/((a+b))`

`=> [(a^2+2ab+b^2) - (a^2+ab+b^2)]/((a+b))`

`=> [ cancel a^2+2ab+cancelb^2- cancela^2-ab-cancelb^2]/((a+b))`

`color(magenta)(=>(ab)/(a+b)`

~Hope this helps! :)