Download Power to a Power Property - Exponents

POWER OF A POWER PROPERTY
(2 )
3
2
→?
Today’s rule, Law #3, will take this introduction problem to the final step in the journey to
completely simplify, negating the need for a calculator.
?
WARM UP PROBLEMS: A)
(a )
m n
Law #3:
( )
x 2 ⋅ x 2 ⋅ x 2 → _____
⇒a
B) x 2
3
( )
C) 2 3
→ _____
2
→2
⇒ _______
D) To find a power to another power
(exponent to another exponent outside a
grouping symbol), you _______________
the exponents.
mn
NOTE #1: A spin-off (or “Offshoot” or “Sub-Series”) is a “new organization or entity formed by a split
from a larger one, such as a television series, based on a pre-existing one, or as a new company formed
from a university research group.” Law #3 can be thought of as a spin-off of Law #1 (see Warm-up
problems A and B).
?
(y )
3
D)
2
( )
E) c 5
→ _______
4
( )
F) 33
→ _______
4
→ 3 ⇒ _________
NOTE #2: For problems like example G where an addition or subtraction symbol are introduced inside the
grouping symbol, be EXTREMELY careful not to try and combine unlike terms inside.
NOTE #3: Another common mistake is multiplying every exponent inside the grouping symbol by the
exponent outside of it when addition or subtraction of bases is present. In example G, the first step is NOT
[ x 3 + 2 3 ] 4 and then misuse Law #3 again to get x12 + 212 !!!
- In example G, the first step also is NOT [(2 x) 3 ]4 (which is mathematically illegal anyway.) The Laws of
Exponents are intended for the multiplication of terms, not addition or subtraction of terms!
?
?
?
[(x + 2) ]
3 4
G)
[( y − 3) ]
3 2
H)
→ ( x + 2)
( )
I) a 5
→ ( y − 3)
3
→ ______
( )
J) 4 2
4
→4
⇒ _____
NOTE #4: In mathematics, the “dash” is thought of as either a 1) Negative sign 2) Subtraction sign and/or
3) “Opposite of” sign. Which of these three is it? It depends upon what you are looking at. If the dash is
within the parentheses, count it with Law #3. For example:
[(− 2) ]
[(− 2) ]
[− (2) ]
[
4
3
is different than − (2 )
4
3
→ [16] = 4,096
3
3
[
K) (− 3)
3
]
2 3
→[
Watch how the two differ:
...Here, the dash is thought of as “Negative 2 to the 4th power.”
→ [− 16] = −4,096
th
the 4 power is.”
4
].
4 3
...Here, the dash is thought of as “The opposite of what the result of 2 to
] 3 ⇒ ______
L)
[(− x ) ]
4 2
→[
] 2 ⇒ ______
NOTE #5: If faced with a problem which combines Laws we have discussed so far, consider the Order of
Operations (P.E.M.D.A.S.). Also, Law #1 may help eliminate bases in the simplifying process, so use first
(although applying later Laws will still generate the same final answer).
?
?
[(x ⋅ x ) ] → [(x ) ]
3
M)
2
7 5
5 2
→ [x
] 2 ⇒ ________