Question

If `9x^2 + 1/(4x^2) = 13,` find the value of `27x^3 + 1/(8x^3)`

Answer 1

Given: `9x^2+1/(4x^2)=13,`

Let’s express it in terms of a binomial:

Here, a = 3x, b = `1/(2x),`

Then, 

a² = 9x², b² = `1/(4x^2)`

So, `(3x)^2 + (1/(2x))^2 = 13`

∴ a² + b² = 13

Finding a + b:

(a + b)² = a² + b² + 2ab

(a + b)² = 13 + 2`(3x)(1/(2x))`

(a + b)² = 13 + 3

(a + b)² = 16

a + b = `+-sqrt16`

∴ a + b = ±4

Using the identity,

a³ + b³ = (a + b)³ − 3ab(a + b)

a³ + b³ = (4)³ − 3`(3x)(1/(2x))`(4)

a³ + b³ = 64 − 3`(3/2)(4)`

a³ + b³ = 64 − `(9/2xx4)`

a³ + b³ = 64 − `(36/2)`

a³ + b³ = 64 − 18

∴ a³ + b³ = 46

Hence, `27x^3 + 1/(8x^3) = a^3 + b^3 = 46`