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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 ,
since he made errors in calculations.
Below I copy-pasted his solution and made corrections in it.
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 <<<---=== : should be x ≈ - 1.098612289/1.609437912 ≈ -0.68261.
x ≈ -0.911 <<<---=== associate correction x ≈ -0.68261.
Therefore, x ≈ -0.911. <<<---=== answer. The CORRECT ANSWER is x ≈ -0.68261.
CHECK. (1/9)^(1/(-0.68261)) + (1/3)^(1/(-0.68261)) = I used Excel to calculate = 29.99951,
which is a good precision.
Solved .
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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.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI 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, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, too.