4 - 1/5
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How do I calculate 4^1.5 without using a calculator?
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John Lind, BS Physics & Mathematics, Arizona State University (1976)
Answered August 17, 2019
Simple enough for me to do in my head.
Has two answers: 8, and -8
The number, 1.5 is the same as 3/2 (three halves). A fractional power is a root; i.e. x^(1/2) or x^(0.5) is, by definition, the square root of x.
4^1.5 is 4^(3/2), the square root of four cubed. Doesn’t matter if you take the square root first or do the cube operation first.
4^(1/2) = 2, and -2
2^3 = 8, and (-2)^3 = -8
Doing the other operation first:
4^3 = 64
64^(1/2) = 8, and -8
This has two answers due to the square root, with the sign carrying through when the result of that is cubed: 8 and -8, or emerging after cubing first and then taking hte square root. Working the problem in reverse using the two answers:
8^(1/3) = 2, and 2^2 = 4, or
8^2 = 64, and 64^(1/3) = 4
-8(1/3) = -2, and -2^2 = 4, or
-8^2 = 64, and 64^(1/3) = 4