3^(-9n) = 3^(-1/2)
-9n = -1/2
n = 1/18
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Huh!? say: (1 / 27)^(3n + 1) = √(3) / 81
(1 / 27)^(3n) * (1 / 27)^(1) = √(3) / 81
Multiply both sides of the equation by 27:
(1 / 27)^(3n) = √(3) * (27 / 81)
(1/ 27)^(3n) = √(3) / 3
Take √(3) / 3, and multiply by √(3) / √(3) which is equivalent to 1.
√(3) / 3 * [√(3) / √(3)]
= 3 / 3(√3)
= 1 / (√3), and that comes out to √(1 / 3) or (1 / 3)^(1 / 2) # This was hiding in the expression. The textbook writers decided to take the square root out of the denominator!
(1 / 27)^(3n) = (1 / 3)^(1 / 2)
Invert both sides of the equation: Raise to the negative 1th power.
[(1 / 27)^(3n)]^(-1) = [(1 / 3)^(1 / 2)]^(-1)
[(1 / 27)^(-1)]^(3n) = [(1 / 3)^(-1)]^(1/2)
(27 / 1)^(3n) = (3 / 1)^(1 / 2)
(3^3)^(3n) = 3^(1 / 2)
3^[(3 * 3n)] = 3^(1 / 2)
The bases of both exponents are the same so equate exponents:
9n = 1 / 2
Divide both sides of the equation by 9, and :
n = 1 / 18
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katie say: i got -19683+343√3 does this sound right?
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Steve A say: (1/27)^(3n+1) = sqrt(3) /81
(3^(-3))^(3n+1) = 3^(1/2)3^(-4)
3^(-9n +3) = 3^(-3.5)
-9n + 3 = -3.5
6.5 = 9n
6.5/9 = n
13/18 = n
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dylan say: Add brackets is it
(1 / 27)^3*(n+1)
(1 / 27)^3*n+1
(1 / 27)^(3n)+1
(1 / 27)^(3n+1)
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Cecile say: i got -19683+343√3 does this sound right?
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