Multiplying Powers
For any rational number n, and for all whole numbers a and b, (na)(nb) = na+b
Illustrative example 1
a2 · a4 = (a · a)(a · a · a · a)
= a2 + 4
= a6
Illustrative example 2
3x3y2 · 5xy3 = 3 · 5 · x3 · x · y2 · y3
= 15x3 + 1y2 + 3
= 15x4y5
Raising a Power to a Power
For any rational number n, and any whole numbers a and b,
(am)n = amn
Illustrative example 1
(xy)3 = (xy)(xy)(xy)
= (x · x · x)(y · y · y)
= x3y3
Illustrative example 2
(4x3y2)2 = (4x3y2)(4x3y2)
= (4·4)(x3 · x3)(y2 · y2) or 4(1)(2)x(3)(2)y(2)(2)
= 42x6y4 or 42x6y4
= 16x6y4 or 16x6y4
Dividing powers
For any rational number a except 0, and for all whole numbers m and n,
am = am-n
an
Definition of a Negative Exponent
For any rational number a except 0, and for all whole numbers m, a-m = .
For any rational number a except 0, a0 = 1.
1. x6 = x · x · x · x · x · x = x6-2 = x4
x2 x · x
2. 12a5b6c3 = 2 · 2 · 3a5b6c3 = 2231-1a5-3b6-4c3-1 = 4a2b2c2
3a3b4c 3a3b4c
1. x3 = x · x · x = 1
x6 x · x · x · x · x · x x3
3. -10m6n4 = -2 · 5 m · m · m · m · m · m · n · n · n · n_____
5m7n6 5 m · m · m · m · m · m · m · n · n · n · n · n · n
= -2 = -2_
m · n · n mn2
4. 8a5b4 = 2 · 2 · 2 · a · a · a · a · a · b · b · b · b = 2(1)(1) = 2
4a5b4 2 · 2 · a · a · a · a · a · b · b · b · b
or = 23-2a5 – 5b4 – 4
= 21a0b0 (Definition 2)
= 2 · 1 · 1
= 2