Exponentiation
See what happens when you use exponentiation. You will need a scientific calculator.
Students may work individually or in groups.
Task
Sort the following twelve numbers by size. Clearly indicate which ones are equal. In order to make the comparisons, compute the decimal approximations of these numbers and use scientific notation.
(2^3)^5, (3^2)^5, (2^5)^3, (5^2)^3, (3^5)^2, (5^3)^2
2^(3^5), 3^(2^5), 2^(5^3), 5^(2^3), 3^(5^2), 5^(3^2)
Solution
(5^2)^3 = 15625
(5^3)^2 = 15625
(2^3)^5 = 32768
(2^5)^3 = 32768
(3^2)^5 = 59049
(3^5)^2 = 59049
5^(2^3) = 390625
5^(3^2) = 1953125
3^(5^2) = 8.472886094 E11
3^(2^5) = 1.853020189 E15
2^(5^3) = 4.253529587 E37
2^(3^5) = 1.413477652 E73
So what is going on?
1. Both the order and the grouping of numbers matters.
In algebraic jargon this means that exponentiation is neither commutative nor associative.
2. We see here that for the numbers x, y, and z above, (x^y)^z = (x^z)^y.
Check that in all three cases above (x^y)^z = x^(y*z).
5^6 = 15625, 2^15 = 32768, and 3^10 = 59049.
These two equalities hold for all x, y and z.