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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\def\arraystretch{1.25} 2^3\cdot 2^2=2^{3+2}\\ \begin{array}{r l} 2^3\cdot 2^2 &= (2\cdot 2\cdot 2)\cdot (2\cdot 2)\\ &=2\cdot 2\cdot 2\cdot 2\cdot 2 \\ &=2^5 \\ &=2^{3+2} \end{array}

axay=ax+ya^x\cdot a^y=a^{x+y}

Multiplication with the same base aa

2322=2322322=222  22=21=232\frac{2^3}{2^2}=2^{3-2}\\ \frac{2^3}{2^2}=\frac{2\cdot 2\cdot 2\;}{2\cdot 2} = 2^1 = 2^{3-2}

axay=axy\,\,\frac{a^x}{a^y}=a^{x-y}

Division with the same base aa

2333=(23)32333=(222)(333)=(23)(23)(23)=(23)3\def\arraystretch{1.25} 2^3\cdot 3^3=\left(2\cdot 3\right)^3\\ \begin{array}{r l} 2^3\cdot 3^3&= ( 2\cdot 2\cdot 2 ) \cdot ( 3\cdot 3\cdot 3 )\\ &= (2 \cdot 3) \cdot (2 \cdot 3) \cdot (2 \cdot 3)\\ &= (2 \cdot 3)^3 \end{array}

    axbx=(ab)x\;\;a^x\cdot b^x=\left(a\cdot b\right)^x

Multiplication with the same exponent xx

2333=(23)32333=222333=232323=(23)3\def\arraystretch{1.25} \frac{2^3}{3^3}=\left(\frac 2 3\right)^3\\ \begin{array}{r l} \frac{2^3}{3^3}&= \frac{2\cdot 2\cdot 2}{3\cdot 3\cdot 3}\\ &= \frac 2 3 \cdot \frac 2 3 \cdot \frac 2 3\\ &= \left(\frac 2 3\right)^3 \end{array}

axbx=(ab)x\frac{a^x}{b^x}=\left(\frac ab\right)^x

Division with the same exponent xx

(23)2=232(23)2=(222)2=(222)(222)=222222=26=232\def\arraystretch{1.25} \left(2^3\right)^2=2^{3\cdot 2}\\ \begin{array}{r l} \left(2^3\right)^2&= ( 2 \cdot 2 \cdot 2 )^2 \\ &= ( 2 \cdot 2 \cdot 2 ) \cdot ( 2 \cdot 2 \cdot 2 )\\ &= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \\ &= 2^6 = 2^{3\cdot 2} \end{array}

    (ax)y=axy\;\;\left(a^x\right)^y=a^{x\cdot y}

Multiple Powers

Common Special Cases

Example

General form

Description

(2)6=26=(2)2×(2)2×(2)2=22×22×22=26\left(-2\right)^6=2^6\\=(-2)^2\times(-2)^2\times(-2)^2\\=2^2\times2^2\times2^2\\= 2^6

(a)x=ax\left(-a\right)^x=a^x

Negative base and even exponent

(2)5=(25)=(2)2×(2)2×(2)=22×22×(2)=(25)\left(-2\right)^5=-(2^5)\\=(-2)^2\times(-2)^2\times(-2)\\=2^2\times2^2\times(-2)\\= -(2^5)

(a)x=(ax)\left(-a\right)^x=-(a^x)

Negative base and odd exponent

20=122=22=22121=2=2120=12^0=1\\2^2 = 2 \cdot 2 = 2 \cdot 2 \cdot 1 \\ 2^1= 2 = 2 \cdot 1 \\ 2^0= 1

a0=1a^0=1

Zero in the exponent with base a0a\ne0

23=1231=20=23+(3)=2323For this to be 1, we need23=123since23123=2323=12^{-3}=\frac{1}{2^3}\\1=2^0=2^{3+(-3)}\\=2^3\cdot 2^{-3}\\\text{For this to be 1, we need}\\ 2^{-3}=\frac1{2^3}\\\text{since}\\2^3\cdot\frac1{2^3}=\frac{2^3}{2^3}=1

ax=1axa^{-x}=\frac{1}{a^x}

Negative exponent

213=232=21=2133=(213)3For this to be 2, we need213=23since(23)3=22^\frac 1 3 =\sqrt[3]2\\2=2^1=2^{\frac13 \cdot 3}\\=\left(2^{\frac13}\right)^3\\ \text{For this to be 2, we need}\\2^{\frac13}=\sqrt[3]2\\ \text{since}\\ \left(\sqrt[3]{2}\right)^3=2

a1n=ana^{\frac{1}{n}}=\sqrt[n]{a}

Unit fractions in the exponent

223=223=2213=(22)13=2232^\frac 23=\sqrt[3]{2^2}\\=2^{2 \cdot \frac13} = \left(2^2\right)^{\frac13}\\=\sqrt[3]{2^2}

amn=amna^{\frac{m}{n}}=\sqrt[n]{a^m}

General fractions in the exponent

Exercises


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