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Power Laws

Power Laws show you how powers act when you multiply, divide or take them to further powers.

Example

General form

Description

2322=23+22322=(222)(22)=22222=25=23+2

axay=ax+y

Multiplication with the same base a

2322=2322322=22222=21=232

axay=axy

Division with the same base a

2333=(23)32333=(222)(333)=(23)(23)(23)=(23)3

axbx=(ab)x

Multiplication with the same exponent x

2333=(23)32333=222333=232323=(23)3

axbx=(ab)x

Division with the same exponent x

(23)2=232(23)2=(222)2=(222)(222)=222222=26=232

(ax)y=axy

Multiple Powers

Common Special Cases

Example

General form

Description

(2)6=26=(2)2×(2)2×(2)2=22×22×22=26

(a)x=ax

Negative base and even exponent

(2)5=(25)=(2)2×(2)2×(2)=22×22×(2)=(25)

(a)x=(ax)

Negative base and odd exponent

20=122=22=22121=2=2120=1

a0=1

Zero in the exponent with base a0

23=1231=20=23+(3)=2323For this to be 1, we need23=123since23123=2323=1

ax=1ax

Negative exponent

213=232=21=2133=(213)3For this to be 2, we need213=23since(23)3=2

a1n=an

Unit fractions in the exponent

223=223=2213=(22)13=223

amn=amn

General fractions in the exponent

Exercises

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