# HamiltonHardison’s Exploration of

Thanks to Katie H., Madelyn K., and Kenny M. for being creative and asking the questions that created this exploration.

While exploring properties of positive integer exponents in an algebra II class, a clever group of students noticed that if you took a number, lets say 99, and raised it to the second power, you got a large number, (in this case 9801).  By squaring the answer, a larger number resulted (96059601).  By repeating the process over and over, the numbers got bigger and bigger until the calculator returned an “overflow” error.

One student asked:  What would happen if you raised it to a smaller power, like 1?

The students quickly discovered that repeatedly raising a number to the first power is not very interesting.
Another student asked:  What would happen if you raised it to a power like 1.5?

 99 985.038 30915.7 5.43587e+06 1.26737e+10

The students were quickly convinced that the same overflow error would result.

An interesting learning opportunity presented itself when a student asked, “What if you raised 10 to a number like .1?”

After exploring 10 to the 1/10th power in the calculator, the students were quickly convinced that, after enough iterations, the result of the operations was one.

Very impressed with the students thinking, I suggested:  What if we try a number like 99 and raise it repeatedly to the .99 power?

The TI-84 was not able to convince the students of a numerical result over the course of more than 10 minutes of key punching (and after having swapped the calculator between 3 different students).

As they punched and punched away on the calculator, I made a simple excel spreadsheet.  The first 200 iterations are shown below.

 99 42.8789 21.6284 12.3573 7.81795 5.3761 3.9578 3.08079 2.51004 2.12276 94.5538 41.2973 20.9736 12.0505 7.65883 5.28643 3.90373 3.04632 2.48704 2.10684 90.3487 39.7889 20.345 11.7543 7.50448 5.19913 3.85092 3.01257 2.46448 2.0912 86.37 38.3499 19.7411 11.4682 7.35474 5.11413 3.79935 2.97953 2.44236 2.07583 82.6036 36.9766 19.161 11.1918 7.20944 5.03134 3.74897 2.94718 2.42064 2.06072 79.0368 35.6654 18.6035 10.9247 7.06843 4.9507 3.69975 2.9155 2.39934 2.04588 75.6573 34.4132 18.0675 10.6666 6.93154 4.87214 3.65167 2.88447 2.37843 2.03129 72.454 33.2168 17.5521 10.4171 6.79863 4.7956 3.60468 2.85407 2.35791 2.01694 69.4163 32.0733 17.0563 10.1758 6.66956 4.72101 3.55875 2.8243 2.33777 2.00284 66.5345 30.9801 16.5793 9.94243 6.54419 4.6483 3.51386 2.79512 2.31801 1.98898 63.7994 29.9345 16.1202 9.71667 6.4224 4.57742 3.46998 2.76654 2.2986 1.97535 61.2024 28.9341 15.6783 9.49822 6.30406 4.50832 3.42707 2.73853 2.27955 1.96195 58.7355 27.9767 15.2526 9.28679 6.18905 4.44094 3.38512 2.71108 2.26084 1.94877 56.3913 27.06 14.8426 9.08212 6.07726 4.37522 3.34409 2.68418 2.24247 1.93581 54.1626 26.1821 14.4476 8.88393 5.96858 4.31112 3.30397 2.6578 2.22444 1.92306 52.0431 25.3411 14.0669 8.69199 5.8629 4.24859 3.26472 2.63195 2.20672 1.91053 50.0264 24.535 13.6999 8.50605 5.76011 4.18757 3.22632 2.6066 2.18933 1.8982 48.1069 23.7623 13.3459 8.32589 5.66013 4.12802 3.18874 2.58175 2.17224 1.88607 46.2791 23.0213 13.0046 8.15129 5.56286 4.06991 3.15198 2.55738 2.15545 1.87415 44.538 22.3105 12.6752 7.98205 5.46821 4.01318 3.116 2.53348 2.13896 1.86241

By cell 1000, excel gives 1.000200401, which is close to 1… But how could they be certain that the number didn’t go below one?

We used Excel to plot a graph…

One student said:  Is that one of those “asym…things?”

While we had not yet introduced the idea of rational exponents, exponential functions, and had barely mentioned the word asymptote, the students were able to discover a lot of mathematics by using technology to develop mathematical thinking and questions.

Algebraically, we have the function  (,which is shown above at right from graphing calculator).

I thought I was going off on a tangent during class (as I didn’t know where the exploration was going); it turns out it was just a secant, as I caught back up to the lesson of the day after a slight detour:  rational exponents.

Send me an e-mail

Returnto Hamilton Hardison's Page