Question 1008619
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9x^(2/3)-26x^(1/3)-3=0
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The standard method solving equations like this is as follows.

Introduce new variable 

y = x^(1/3) = {{{x^(1/3)}}}       (1).

Then your equation takes the form

{{{9y^2}}} - {{{26y}}} - {{{3}}} = {{{0}}}.   (2)

It is a quadratic equation for y. Solve it using the quadratic formula:

{{{y[1,2]}}} = {{{(26 +- sqrt(26^2 + 4*9*3))/18}}} = {{{(26 +- sqrt(784))/18}}} = {{{(26 +- 28)/18}}}.

Thus the equation (2) has two solutions: {{{y[1]}}} = 3;   {{{y[2]}}} = {{{-1/9}}}.

Now, to find x, you need solve two equations ( recall (1) ):

1) {{{x^(1/3)}}} = 3. To solve it, raise both side to the degree 3: x = {{{3^3}}} = 27.

2) {{{x^(1/3)}}} = {{{-1/9}}}. To solve it, raise both side to the degree 3: x = {{{(-1/9)^3}}} = {{{-1/729}}}.

<U>Answer</U>. There are two solutions of the original equation: x = 27 and x = {{{-1/729}}}.
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