Simplify the following:
5/6^(80 (5/6-1)/10-1)
80 (5/6-1)/10 = (80 (5/6-1))/10:
5/6^((80 (5/6-1))/10-1)
Put 5/6-1 over the common denominator 6. 5/6-1 = 5/6-(6)/6:
5/6^((80 5/6-(6)/6)/10-1)
5/6-(6)/6 = (5-6)/6:
5/6^((80 (5-6)/6)/10-1)
5-6 = -1:
5/6^((80×-1/6)/10-1)
80×(-1)/6 = (80 (-1))/6:
5/6^((-80)/6/10-1)
((80 (-1))/6)/10 = (80 (-1))/(6×10):
5/6^((-80)/(6×10)-1)
6×10 = 60:
5/6^((-80)/60-1)
The gcd of -80 and 60 is 20, so (-80)/60 = (20 (-4))/(20×3) = 20/20×(-4)/3 = (-4)/3:
5/6^(-4/3-1)
Put (-4)/3-1 over the common denominator 3. (-4)/3-1 = (-4)/3-(3)/3:
5/6^-4/3-(3)/3
(-4)/3-(3)/3 = (-4-3)/3:
5/6^(-4-3)/3
-4-3 = -7:
5/6^(-7/3)
6^(-7/3) = 6^(2/3-(9)/3) = 6^(-9/3)×6^(2/3):
5/6^(-9/3)×6^(2/3)
9/3 = (3 (-3))/3 = 3:
5/(6^(-3)×6^(2/3))
6^3 = 6×6^2:
5/(1/(6×6^2) 6^(2/3))
6^2 = 36:
5/((6^(2/3))/(6×36))
6×36 = 216:
5/((6^(2/3))/(216))
Multiply the numerator of 5/((6^(2/3))/(216)) by the reciprocal of the denominator. 5/((6^(2/3))/(216)) = (5×216)/6^(2/3):
(5×216)/6^(2/3)
5×216 = 1080:
1080/6^(2/3)
Rationalize the denominator. 1080/6^(2/3) = 1080/6^(2/3)×(6^(1/3))/(6^(1/3)) = (1080 6^(1/3))/(6):
(1080 6^(1/3))/(6)
6 | | 1 | 8 | 0
| 1 | 0 | 8 | 0
- | | 6 | |
| | 4 | 8 |
| - | 4 | 8 |
| | | | 0
| | | - | 0
| | | | 0:
Answer: | Log[180 6^(1/3)]=2.5146562...
