Need help with this algebra problem

[gamaz321]

I got this problem where I could do quite a few steps. However, I am stuck at one point where I cannot advance. I appreciate any help on this.

 

[Steven G]

(p+q)^2/3-(p-q)^2/3 is the difference of squares.
Can you go with that?

 

[gamaz321]

Thank you for your advice. The problem is how to get the value of a^1/3 + b^1/3. Unless I find this value we cannot calculate (p + q) ^2/3 even with your clue. How does one find the value of a^1/3 + b^1/3?

 

[JeffM]

Thank you for your advice. The problem is how to get the value of a^1/3 + b^1/3. Unless I find this value we cannot calculate (p + q) ^2/3 even with your clue. How does one find the value of a^1/3 + b^1/3?

Your y substitution was a good idea. Why did you abandon it before calculating y?

Moreover,

[math] p = a + 3a^{1/3}b^{2/3} \text { and } q = b + 3a^{2/3}b^{1/3} \implies \\ p + q = a + 3a^{1/3}b^{2/3} + b + 3a^{2/3} = a^{3/3} + 3a^{2/3}b^{1/3} + 3a^{1/3}b^{2/3} + b^{3/3} = (a^{1/3} + b^{1/3})^3.\\ p - q = \text {WHAT?} [/math]

 

[gamaz321]

Thanks for pointing out the fact that I did abandon the calculation of a and b. However, I went ahead and calculated a and b. Unfortunately, it did not get any meaningful results. I had to use a calculator to calculate the value of a which should not be the case. So I am not sure how to proceed from here.