SOLUTION: How do you find x^2 if x satisfies (x+9)^(1/3)-(x-9)^(1/3)=3

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Question 212297: How do you find x^2 if x satisfies (x+9)^(1/3)-(x-9)^(1/3)=3
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
How do you find x%5E2 if x satisfies
1. %28x%2B9%29%5E%281%2F3%29-%28x-9%29%5E%281%2F3%29=3

Let A+=+x%2B9 and B=x-9

2. A%5E%281%2F3%29-B%5E%281%2F3%29=3

Cube both sides:

   A-3A%5E%282%2F3%29B%5E%281%2F3%29%2B3A%5E%281%2F3%29B%5E%282%2F3%29-B=27

Group the first and last terms on the left together

3. %28A-B%29-3A%5E%282%2F3%29B%5E%281%2F3%29%2B3A%5E%281%2F3%29B%5E%282%2F3%29=27

Notice that A-B=%28x%2B9%29-%28x-9%29=x%2B9-x%2B9=18
Substitute 18 for A-B

18-3A%5E%282%2F3%29B%5E%281%2F3%29%2B3A%5E%281%2F3%29B%5E%282%2F3%29=27

Subtract 18 from both sides:

-3A%5E%282%2F3%29B%5E%281%2F3%29%2B3A%5E%281%2F3%29B%5E%282%2F3%29=9

Divide through by -3

A%5E%282%2F3%29B%5E%281%2F3%29-A%5E%281%2F3%29B%5E%282%2F3%29=-3

Factor out A%5E%281%2F3%29B%5E%281%2F3%29 on the left

A%5E%281%2F3%29B%5E%281%2F3%29%28A%5E%281%2F3%29-B%5E%281%2F3%29%29=-3

Using equation 2 above, the parenthetical expression
equals 3

A%5E%281%2F3%29B%5E%281%2F3%29%283%29=-3

Divide both sides by 3

A%5E%281%2F3%29B%5E%281%2F3%29=-1

Cube both sides:

%28A%5E%281%2F3%29B%5E%281%2F3%29%29%5E3=%28-1%29%5E3

AB=-1

Substitute A+=+x%2B9 and B=x-9, 

%28x%2B9%29%28x-9%29=-1

x%5E2-81=-1

Add 81 to both sides:

x%5E2=80

That's what you wanted.

Edwin