Simplify each of the following powers of i. i^32

  i^32 = 1

Step-by-step explanation:

For i = √(-1), i^2 = -1 and i^4 = (-1)^2 = 1.

We know that ...

  i^32 = (i^4)^8

Substituting for i^4, we have ...

  i^32 = 1^8 = 1

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.