1. Multiply–Add Rule: AnAk = An+k
Divide–Subtract Rule: An
Ak
Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk
Exponents
= An – k
Exponent–Rules
0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
1/n - Power Rule: A1/n = A , the nth root of A.
n
(divide–subtract)
(divide–subtract)
2. Special Exponents
Example B.
c. 641/3 = 64 = 4
3
b. 81/3 = 8 = 2
3
a. 641/2 = 64 = 8
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
Example C. Find the root first, then raise the root to
the numerator–power.
The fractional exponent
A
k
n±
(A ) kn ±1
is
1. take the nth root of A
2. then raise the
root to ±k power
c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-3 = 1/23 = 1/84
b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9
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1/n - Power Rule: A1/n = A , the nth root of A.
n
3. Fractional Powers
x*(x1/3y3/2)2
x–1/2y2/3 =
x*x2/3y3
x–1/2y2/3 = x–1/2y2/3 =
x5/3y3
x5/3 – (–1/2) y3 – 2/3
= x13/6 y7/3
Example D. Simplify by combining the exponents.
power–multiply rule
1/3*2 3/2*2
multiply–add rule
1 + 2/3 divide–subtract rule
To write a radical in fractional exponent form:
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
a2 a = a2/3a1/4 = a11/12 = a11
3 4 12
c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).
a. 53 or (5 )3 = 53/2 b. 9a2 = (9a2)1/2 = 3a
d. Multiply
Use the exponent-rules to collect exponents.
4. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8
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c. 30.4 = 32/5 = (3)2 ≈ 1.55 (by calculator)
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