Question

What is the value of ?

`x^3-3b^(2/3)x+9a` , if x = (2a+
`sqrt(4a^2-b^2))^(1/3)` + (2a-`sqrt(4a^2-b^2))^(1/3)` ?

options :-
1. 12a
2. 13a
3. 14a
4. none of these

please answer with full explanation

Answer 1

Option `4 -> "None of these"`

Explanation:

Follow this 3 easy steps, it's not that hard as it seems..

`x^3 - 3b^(2/3) x + 9a`

Where `x = (2a + sqrt(4a^2 - b^2))^(1/3) + (2a - sqrt(4a^2 - b^2))^(1/3)`

Step 1 `->` Substitute the value of x into the main equation..

`color(red)x^3 - 3b^(2/3) color(red)(x) + 9a`

`color(red) [[(2a + sqrt(4a^2 - b^2))^(1/3) + (2a - sqrt(4a^2 - b^2))^(1/3)]]^3 - 3b^(2/3) color(red)[[(2a + sqrt(4a^2 - b^2))^(1/3) + (2a - sqrt(4a^2 - b^2))^(1/3)] + 9a`

Step 2 `->` Eliminating the powers..

`[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(1/3 xx cancel3) - 3b^(2/3) [(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(1/3) + 9a`

`[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))] - 3b^(2/3) [(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(1/3) + 9a`

`[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))] - 3b[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^((1/3) xx (2/3)) + 9a`

`[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))] - 3b[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(2/9) + 9a`

Step 3 `->` Collecting like terms..

`(2a + 2a) + (sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2)) - 3b[(2a + 2a) + (sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2))]^(2/9) + 9a`

`4a + cancel(sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2)) - 3b[4a + cancel(sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2))]^(2/9) + 9a`

`rArr` `4a + 0 - 3b[4a + 0]^(2/9) + 9a`

`rArr` `4a - 3b[4a]^(2/9) + 9a`

`rArr` `4a + 9a - (3b xx 4a)^(2/9)`

`rArr` `13a - 12ab^(2/9) -> Answer`

Hence Option 4 is the suitable Answer..