Question

If x>0 and x2+1/9x2=25/36 find x3+1/27x3

Answer 1

Given Equation is x^2 + 1/9x^2 = 25/36.

We know that (x + 1/3x)^2 = x^2 + 1/9x^2 + 2 * x * 1/3x

                                            = x^2 + 1/9x^2 + 2/3

                                           = 25/36 + 2/3

                                           = 25 + 2* 12/36

                                           = 25 + 24/36

                                           = 49/36

                                x + 1/3x = 7/6.

Now,

On cubing both sides, we get

(x + 1/3x)^3 = (7/6)^3

x^3 + 1/27x^3 + 3 * x^2 * 1/3x + 3 * x * 1/9x^2 = 343/216

x^3 + 1/27x^3 + x + 1/3x = 343/216

x^3 + 1/27x^3 + 7/6 = 343/216

x^3 + 1/27x^3 = 343/216 - 7/6

x^3 + 1/27x^3 = (343 - 7 * 36)/(216)

x^3 + 1/27x^3 = 91/216.

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