Question 1209741
.
If 1/9^(1/x) + 1/3^(1/x) = 30, 
find x.
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        The final answer in the post by @CPhill is  {{{highlight(highlight(incorrect))}}},
        since he made errors in calculations.


        Below I copy-pasted his solution and made corrections in it.



<pre>
Here's how to solve the equation 1/9^(1/x) + 1/3^(1/x) = 30:

1. **Rewrite with a common base:** Notice that 9 = 3². We can rewrite the first term as:

   1/9^(1/x) = 1/(3²)^(1/x) = 1/3^(2/x) = (1/3^(1/x))²

2. **Substitute:** Let y = 1/3^(1/x).  The equation becomes:

   y² + y = 30

3. **Rearrange:**

   y² + y - 30 = 0

4. **Factor:**

   (y + 6)(y - 5) = 0

5. **Solve for y:**

   y = -6 or y = 5

6. **Consider the valid solution:** Since y = 1/3^(1/x), y must be positive. Therefore, y = -6 is not a valid solution.  We are left with:

   y = 5

7. **Substitute back:**

   1/3^(1/x) = 5

8. **Rewrite:**

   3^(-1/x) = 5

9. **Take the logarithm of both sides (any base will work, but natural log is common):**

   ln(3^(-1/x)) = ln(5)

10. **Use logarithm power rule:**

   (-1/x) * ln(3) = ln(5)

11. **Solve for x:**

   -1/x = ln(5) / ln(3)
   1/x = -ln(5) / ln(3)
   x = -ln(3) / ln(5)

12. **Calculate:**

   x ≈ -1.465 / 1.609   <<<---===  {{{highlight(highlight(error))}}}:  should be    x ≈ - 1.098612289/1.609437912 ≈ -0.68261.

   x ≈ -0.911           <<<---===  associate correction x ≈ -0.68261.
 
Therefore, x ≈ -0.911.  <<<---===  {{{highlight(highlight(wrong))}}} answer.  The  <U>CORRECT ANSWER</U>  is  x ≈ -0.68261.


<U>CHECK</U>.  (1/9)^(1/(-0.68261)) + (1/3)^(1/(-0.68261)) = I used Excel to calculate = 29.99951,
                                                           which is a good precision.
</pre>

Solved &nbsp;&nbsp;{{{highlight(highlight(correctly))}}}.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . 



Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.


The artificial intelligence is like a baby now. It is in the experimental stage 
of development and can make mistakes and produce nonsense without any embarrassment.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.



This time, again, &nbsp;it made an error.



Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.


Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental
and can make errors/mistakes.


All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.


Every time, &nbsp;@CPhill embarrassed to tell the truth.

But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.



And the last my comment.


When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, 
until the specialists and experts will check and confirm their correctness.


Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.