Ben Wiltbank shared with us a grid he thinks of that helps him remember how to simplify expressions with powers. I will call it the Wiltbank Grid, though I don't suppose he is the first to share it. It looks like this:

Level 3 | power | is the opposite of | root |

Level 2 | * (multiply) | is the opposite of | / (divide) |

Level 1 | + (plus) | is the opposite of | - (minus) |

Wiltbank's rule is that if there are powers of some number, x, that are connected using the operation at some level in the grid, you can simplify by merging the occurences of x and applying the next lower level operation to the exponents.

Some examples:

x

^{2}*x

^{2}=x

^{(2+2)}Since "+" is one level below "*".

x

^{2}+x

^{2}Can't be simplified, because "+" is the lowest level.

(x

^{2})

^{2}=x

^{(2*2)}Since "*" is one level below "power".

x

^{2}/x

^{2}=x

^{(2-2)}Since "-" is one level below "/".

x

^{2}-x

^{2}Can't be simplified, because "-" is the lowest level.

Now the answers to the quiz:

1. 7

2. +

3. * (times)

4. / (divide)

5. 25

6. 1/3

7. 4

8. ok

9. /

10a. /

10b. ok

11a. ok

11b. ok

11c. /

12. ok

13a. /

13b. ok

13c. ok

14a. ok

14b. /

14c. ok

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