SOLUTION: Let P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81}. Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possib

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Question 1209817: Let
P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81}.
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$
Found 3 solutions by CPhill, Edwin McCravy, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
Let's break down the problem step by step:
1. **Express all terms with the same base:**
* P = 3^(1/3) * (3^2)^(1/9) * (3^3)^(1/27) * (3^4)^(1/81)
* P = 3^(1/3) * 3^(2/9) * 3^(3/27) * 3^(4/81)
2. **Combine the exponents:**
* P = 3^(1/3 + 2/9 + 3/27 + 4/81)
3. **Find a common denominator:**
* The least common denominator is 81.
* P = 3^(27/81 + 18/81 + 9/81 + 4/81)
4. **Add the fractions:**
* P = 3^(58/81)
5. **Express in radical form:**
* P = (3^58)^(1/81)
* P = ⁸¹√(3^58)
6. **Identify a and b:**
* a = 81
* b = 3^58
7. **Calculate a + b:**
* a + b = 81 + 3^58
8. **Calculate 3^58:**
* 3^58 = 4710128697246244834921603770
9. **Calculate a+b**
* a + b = 81 + 4710128697246244834921603770 = 47101286972462448349216037751
Therefore, the smallest possible value of a + b is 81 + 3^58.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
There's that notation again that I have not completely figured out which is
not compatible with the HTML this site is written in.  Maybe this the correct
interpretation.  [If this was posted by a human, I would appreciate it if you
post again, to use standard notation.]

 

a=81, b=358.

So the smallest possible value of 81+358 = 4710128697246244834921603770

Edwin
Answer by ikleyn(52797)   (Show Source): You can put this solution on YOUR website!
.

@CPhill writes in his post 

    8. **Calculate 3^58:**
       * 3^58 = 4710128697246244834921603770


It is totally wrong.

 can not end by the  "0"  digit.

Last digit of      is  9.


So,  the answer in the post by  @CPhill is  INCORRECT.



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