Question 1209741: If 1/9^(1/x) + 1/3^(1/x) = 30,
find x.
Found 3 solutions by CPhill, ikleyn, mccravyedwin:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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
x ≈ -0.911
Therefore, x ≈ -0.911.
Answer by ikleyn(52866)  (Show Source):
You can put this solution on YOUR website! .
If 1/9^(1/x) + 1/3^(1/x) = 30,
find x.
~~~~~~~~~~~~~~~~~~~~~~~~~
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.
Answer by mccravyedwin(408)  (Show Source):
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