Prove x^(1/3)+(2x-3)^(1/3)=(12(x-1))^(1/3)
I am totally stuck. How would I edit the left hand side to look like the right hand side? Is it to do with (A+B)^3 or (A^3+B^3)? I can't seem to see the pattern!
Edit: WAIT! Never mind! I understand now!
Prove x^(1/3)+(2x-3)^(1/3)=(12(x-1))^(1/3)
I am totally stuck. How would I edit the left hand side to look like the right hand side? Is it to do with (A+B)^3 or (A^3+B^3)? I can't seem to see the pattern!
Edit: WAIT! Never mind! I understand now!
Your solution has MAJOR flaws, in that you have only shown the equation is true when x = 1 and x = 3, not for all x.
Also, you can NOT start by writing the equation and working on both sides as if it was true. You are trying to show it is true, which means you can't make the assumption at the beginning (unless you were using a proof by contradiction, in which case you need to assume the OPPOSITE is true).
You have to start with one side and manipulate it until you get to the other side.
I apologize deeply for the ambiguity! The reason I posted this question in the first place is because I was confused about the solution key (the document that I previously attached). From the transition from steps (4) to (5), I was confused as to how (x)^(1/3)+(2x-3)^(1/3) turned into (12(x-1))^(1/3). So I posted the question in mind for someone to maybe help me simplify (x)^(1/3) + (2x-3)^(1/3) to (12(x-1))^(1/3). But as it turns out, the original question explicitly stated that the two terms equal to (12(x-1))^(1/3). So I substituted it back in and solved the problem myself. Therefore, I edited the thread and said "problem solved".
So essentially, the question is actually "Find the value(s) of x that satisfy the equation". But I asked the earlier question to "prove that (x)^(1/3)+(2x-3)^(1/3)=(12(x-1))^(1/3)" because I believed that there was somehow a way to make it work, but I don't believe there is a way; there is only the method of substituting it back into the worked out equation.
Hopefully I explained this clearly... to recap I only initially asked to prove blah + blah = bloop because I assumed there was some way to substitute it into the equation. The actual question is to find x!