DragginMath: Rules of Exponents

The operators raise↑, root√ , and log↓ can be commuted, associated, distributed, and factored in various ways. Although these words are not traditionally used in this context, they are used here because it is hard to see why they are not, at least in a casual sense.

Traditional language places all of these things under the general heading Rules of Exponents.

a↑b = ⅟b√a by dragging a or b sideways.
a↑⅟b = b√a by dragging a or ⅟sideways.
a↑(b+c) = a↑b∗a↑c by dragging + up onto ↑.
a↑(b–c) = a↑b÷a↑c by dragging – up onto ↑.
a↑(b∗c) = (a↑b)↑c by dragging ∗ up onto ↑.
a↑(b÷c) = c√a↑b by dragging ÷ up onto ↑.
(a∗b)↑c = a↑c∗b↑c by dragging ∗ up onto ↑.
(a÷b)↑c = a↑c÷b↑c by dragging ÷ up onto ↑.
(a↑b)↑c = a↑(b∗c) by dragging ↑ up onto ↑.
(a√b)↑c = a√(b↑c) by dragging ↑ up onto ↑.
a↑b∗a↑c = a↑(b+c) by dragging ↑ up onto ↑.
a↑b÷a↑c = a↑(b–c) by dragging a up onto a.
a↑c∗b↑c = (a∗b)↑c by dragging a up onto a.
a↑c÷b↑c = (a÷b)↑c by dragging c up onto c.
a√b = b↑⅟a by dragging ⅟or b sideways.
 ⅟a√b = b↑a by dragging ⅟or b sideways.
a√(b∗c) = a√b∗a√c by dragging ∗ up onto √.
a√(b÷c) = a√b÷a√c by dragging ÷ up onto √.
a√b↑c = (a√b)↑c by dragging ↑ up onto √.
a√b√c = (a∗b)√c by dragging √ up onto √.
(a∗b)√c = a√b√c by dragging ∗ up onto √.
(a÷b)√c = a√c↑b by dragging ÷ up onto √.
a√b∗a√c = a√(b∗c) by dragging a up onto a.
a√b÷a√c = a√(b÷c) by dragging a up onto a.
a√c∗b√c = ⅟(⅟a+⅟b)√c by dragging c up onto c.
a√c÷b√c = ⅟(⅟a–⅟b)√c by dragging c up onto c.
a↓b = ⅟(b↓a) by dragging a or b sideways.
⅟(a↓b) = b↓a by dragging a or b sideways.
a↓(b∗c) = a↓b+a↓c by dragging ∗ up onto ↓.
a↓(b÷c) = a↓b–a↓c by dragging ÷ up onto ↓.
a↓b↑c = a↓b∗c by dragging ↑ up onto ↓.
a↓b√c = a↓c÷b by dragging √ up onto ↓.
a↓b∗c = a↓b↑c by dragging ↓ up onto ∗.
a↓b÷c = a↓c√b by dragging ↓ up onto ÷.
a↓b+a↓c = a↓(b∗c) by dragging a up onto a.
a↓b–a↓c = a↓(b÷c) by dragging a up onto a.

There are so many of these, some might easily have been missed, either when making this list, or when implementing DragginMath. If you know a transformation that DragginMath doesn’t do, please tell us about it.

DragginMath can do things with raise↑, root√, and log↓ that are more complicated than these basic examples. For example, do you remember the different ways of invoking general and specific distribution of multiplication over addition? Those work here, too. Enter a↑(b+c∗d), then drag + up onto ↑. The result is a↑b∗(a↑c)↑d. But if you drag b or ∗ up onto ↑, the result is a↑b∗a↑(cd). The first example seeks out everything that can work in that subtree, but the second example stops where you tell it.

As always, if you want to know if DragginMath can do something, think up an example and try it.