Question

`x^2` plus `1/(9x^2)` is equal to `25/36`. Find the value of `x^3` plus `1/(27x^3)`?

Answer 1

`+-91/216`.

Explanation:

Given that, `x^2+1/(9x^2)=25/36`.

`:. (9x^4+1)/(9x^2)=25/36`.

Cancelling `9` from the `drs.` of both sides, we get,

`(9x^4+1)/x^2=25/4, or, 36x^4+4=25x^2`.

`:. 36x^4-25x^2+4=0`.

`:. ul(36x^4-16x^2)-ul(9x^2+4)=0`.

`:.4x^2(9x^2-4)-1(9x^2-4)=0`.

`:. (9x^2-4)(4x^2-1)=0`.

`:. x^2=4/9, or, x^2=1/4`.

`:. x=+-2/3, or, x=+-1/2`.

Case 1: `x=+-2/3`.

`x=+-2/3 rArr 1/x=+-3/2`.

`:."The Reqd. Value="x^3+1/(27x^3)=+-8/27+1/27(+-27/8)`,

`=+-8/27+-1/8=+-91/216`.

Case 2: `x=+-1/2`.

`"In this Case, the Reqd. Value="+-1/8+1/27(+-8)=+-91/216`.

Altogether, the Reqd. Value`=+-91/216`.

Answer 2

`x^3+1/(27x^3) = +-91/216`

Explanation:

Here's an alternative method that does not require calculating `x`:

Note that:

`(x+1/(3x))^2 = x^2+2/3+1/(9x^2) = 25/36+2/3 =49/36 = (7/6)^2`

So:

`(x+1/(3x)) = +-7/6`

Then:

`+-343/216 = (x+1/(3x))^3`

`color(white)(+-343/216) = x^3+x+1/(3x)+1/(27x^3)`

`color(white)(+-343/216) = x^3+1/(27x^3)+-7/6`

So:

`x^3+1/(27x^3) = +-(343/216-7/6) = +-(343/216-252/216) = +-91/216`