Question 1209570: 3^a + 9^a + 27^a = 39
find a
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52747)   (Show Source):
You can put this solution on YOUR website! .
3^a + 9^a + 27^a = 39
find a
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Function a -->  is a monotonically increasing function of "a".
At a= 1, we have  = 3 + 9 + 27 = 39.
Hence, a= 1 is the unique real solution to the given equation. ANSWER
At this point, the problem is solved completely.
Answer by MathTherapy(10549)  (Show Source):
You can put this solution on YOUR website!
3^a + 9^a + 27^a = 39
= 39
 = 39
 = 39
 = 39 ------ Substituting t for 
 <==== The highest exponent is 3, so there're THREE (3) solutions to this equation.
Using the RATIONAL ROOT theorem, we find that 3 is a ROOT of the equation. Therefore, t = 3, so t - 3 = 0, and t - 3
is a factor. With 3 being a root, we use SYNTHETIC DIVISION or LONG DIVISION of POLYNOMIALS to find the other factor.
The former gives us: 3 | 1 | 1 | 1 | - 39 |
| | 3 | 12 | 39 |
| 1 | 4 | 13 | 0 |
Thus, the other factor is:  , and so,  becomes:  , which gives us:
t - 3 = 0 or 
t = 3 or The DISCRIMINANT for the quadratic is:  .
With the DISCRIMINANT being NEGATIVE (< 0), 2 of the roots to this quadratic are
IMAGINARY/COMPLEX.
Since t = 3, the ONLY REAL root is 3.
t = 3
= 3 ------ Back-substituting for t
As the BASES are equal, so are the EXPONENTS. As such, a = 1.
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