Exponentiation 55
Solutiona. ⎛⎝⎜⎛⎛⎝⎝− ⎞⎠⎟⎞⎞⎠⎠1
823 /Step 1. Rewrite ⎛−
⎝⎛⎛⎛⎛
⎝⎝
⎛⎛⎛⎛ ⎞
⎠
⎞⎞⎞⎞
⎠⎠
(^1) ⎞⎞⎞⎞
8
23 /
using xm//n ()x/nm.
⎛−
⎝
⎛⎛⎛⎛
⎝⎝
⎛⎛⎛⎛ ⎞
⎠
⎞⎞⎞⎞
⎠⎠
⎞⎞⎞⎞ =−⎛
⎝
⎛⎛
⎝⎝
⎞
⎠
⎞⎞⎞⎞
⎠⎠
⎡ ⎞⎞⎞⎞
⎣
⎢
⎡⎡
⎣⎣
⎤
⎦
⎥
⎤⎤
⎦⎦
1
8
1
8
23 13 /^2
⎛/⎡ (^1) ⎞
3 1
Step 2. Find ⎛−
⎝
⎛⎛⎛⎛
⎝⎝
⎛⎛⎛⎛ ⎞
⎠
⎞⎞⎞
⎠⎠
(^1) ⎞⎞⎞⎞
8
13 /
.
⎝⎝⎛⎛⎛⎝⎛⎛⎛⎛⎛⎛− ⎞⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ =−
1
8
1
2
13 /Step 3. Raise −1
2
to the second power.⎡−
⎣⎢
⎡⎡
⎣⎣
⎤
⎦⎥
⎤⎤
⎦⎦
=
1
2
1
4
2Step 4. Review the main results.⎛−
⎝⎛⎛⎛⎛
⎝⎝
⎛⎛⎛⎛ ⎞
⎠
⎞⎞⎞⎞
⎠⎠
⎞⎞⎞⎞ =−⎛
⎝
⎛⎛
⎝⎝
⎞
⎠
⎞⎞⎞⎞
⎠⎠
⎡ ⎞⎞⎞⎞
⎣
⎢
⎡⎡
⎣⎣
⎤
⎦
⎥
⎤⎤
⎦⎦
=−⎡
⎣⎢
⎡⎡
⎣⎣
⎤
⎦⎥
⎤⎤
⎦⎦
=
1
8
1
8
1
2
1
4
23 // 132 2⎡⎛ (^1) ⎞
3 1
b. ()^32 //
Step 1. Rewrite () 3632 / using xxm//n=( nm).
() 3632 / =() 36361212 /^3
Step 2. Find () 3612 /.
() 3612 / = 6
Step 3. Raise 6 to the third power.
63 = 216
Step 4. Review the main results.
()^3 /// ==()(()/^23366 = 216
()^3632 / ≠⋅^363623.^ ()^3632 / =^216 , but
36 ⋅=^3254. Don’t multiply the base by the
exponent!