Simplify the following:
81^(3/2) - 45^(3/2)
45^(3/2) = 45^(2/2 + 1/2) = 45^(2/2)×sqrt(45):
81^(3/2) - 45^(2/2) sqrt(45)
2/2 = 1:
81^(3/2) - 45 sqrt(45)
sqrt(45) = sqrt(3^2×5) = 3 sqrt(5):
81^(3/2) - 45×3 sqrt(5)
45×3 = 135:
81^(3/2) - 135 sqrt(5)
81^(3/2) = 81^(2/2 + 1/2) = 81^(2/2)×sqrt(81):
81^(2/2) sqrt(81) - 135 sqrt(5)
2/2 = 1:
81 sqrt(81) - 135 sqrt(5)
sqrt(81) = sqrt(3^4) = 3^2:
81×3^2 - 135 sqrt(5)
3^2 = 9:
81×9 - 135 sqrt(5)
81×9 = 729:
729 - 135 sqrt(5)
Factor 27 out of 729 - 135 sqrt(5) giving 27 (27 - 5 sqrt(5)):
Answer: |27 (27 - 5 sqrt(5))
