Power Laws

$a^x\cdot a^y=a^{x+y}a^x\backslash cdot\; a^y=a^\{x+y\}$

Multiplication with the same base $aa$

$\frac{2^3}{2^2}=2^{3-2}\frac{2^3}{2^2}=\frac{2\cdot 2\cdot 2}{2\cdot 2}=2^1=2^{3-2}\backslash frac\{2^3\}\{2^2\}=2^\{3-2\}\backslash \backslash \; \backslash frac\{2^3\}\{2^2\}=\backslash frac\{2\backslash cdot\; 2\backslash cdot\; 2\backslash ;\}\{2\backslash cdot\; 2\}\; =\; 2^1\; =\; 2^\{3-2\}$

$\frac{a^x}{a^y}=a^{x-y}\backslash ,\backslash ,\backslash frac\{a^x\}\{a^y\}=a^\{x-y\}$

Division with the same base $aa$

$a^x\cdot b^x={(a\cdot b)}^x\backslash ;\backslash ;a^x\backslash cdot\; b^x=\backslash left(a\backslash cdot\; b\backslash right)^x$

Multiplication with the same exponent $xx$

$\frac{a^x}{b^x}={\left(\frac{a}{b}\right)}^x\backslash frac\{a^x\}\{b^x\}=\backslash left(\backslash frac\; ab\backslash right)^x$

Division with the same exponent $xx$

${(-2)}^6=2^6=(-2{)}^2\times (-2{)}^2\times (-2{)}^2=2^2\times 2^2\times 2^2=2^6\backslash left(-2\backslash right)^6=2^6\backslash \backslash =(-2)^2\backslash times(-2)^2\backslash times(-2)^2\backslash \backslash =2^2\backslash times2^2\backslash times2^2\backslash \backslash =\; 2^6$

${(-a)}^x=a^x\backslash left(-a\backslash right)^x=a^x$

Negative base and even exponent

${(-2)}^5=-(2^5)=(-2{)}^2\times (-2{)}^2\times (-2)=2^2\times 2^2\times (-2)=-(2^5)\backslash left(-2\backslash right)^5=-(2^5)\backslash \backslash =(-2)^2\backslash times(-2)^2\backslash times(-2)\backslash \backslash =2^2\backslash times2^2\backslash times(-2)\backslash \backslash =\; -(2^5)$

${(-a)}^x=-(a^x)\backslash left(-a\backslash right)^x=-(a^x)$

Negative base and odd exponent

$2^0=12^2=2\cdot 2=2\cdot 2\cdot 12^1=2=2\cdot 12^0=12^0=1\backslash \backslash 2^2\; =\; 2\; \backslash cdot\; 2\; =\; 2\; \backslash cdot\; 2\; \backslash cdot\; 1\; \backslash \backslash \; 2^1=\; 2\; =\; 2\; \backslash cdot\; 1\; \backslash \backslash \; 2^0=\; 1$

$a^0=1a^0=1$

Zero in the exponent with base $a\mathrm{\ne }0a\backslash ne0$

$2^{-3}=\frac{1}{2^3}1=2^0=2^{3+(-3)}=2^3\cdot 2^{-3}\text{For this to be 1, we need}{2}^{-3}=\frac{1}{2^3}\text{since}{2}^3\cdot \frac{1}{2^3}=\frac{2^3}{2^3}=12^\{-3\}=\backslash frac\{1\}\{2^3\}\backslash \backslash 1=2^0=2^\{3+(-3)\}\backslash \backslash =2^3\backslash cdot\; 2^\{-3\}\backslash \backslash \backslash text\{For\; this\; to\; be\; 1,\; we\; need\}\backslash \backslash \; 2^\{-3\}=\backslash frac1\{2^3\}\backslash \backslash \backslash text\{since\}\backslash \backslash 2^3\backslash cdot\backslash frac1\{2^3\}=\backslash frac\{2^3\}\{2^3\}=1$

$a^{-x}=\frac{1}{a^x}a^\{-x\}=\backslash frac\{1\}\{a^x\}$

Negative exponent

$2^{\frac{1}{3}}=\sqrt[3]{2}2=2^1=2^{\frac{1}{3}\cdot 3}={\left(2^{\frac{1}{3}}\right)}^3\text{For this to be 2, we need}{2}^{\frac{1}{3}}=\sqrt[3]{2}\text{since}{\left(\sqrt[3]{2}\right)}^3=22^\backslash frac\; 1\; 3\; =\backslash sqrt[3]2\backslash \backslash 2=2^1=2^\{\backslash frac13\; \backslash cdot\; 3\}\backslash \backslash =\backslash left(2^\{\backslash frac13\}\backslash right)^3\backslash \backslash \; \backslash text\{For\; this\; to\; be\; 2,\; we\; need\}\backslash \backslash 2^\{\backslash frac13\}=\backslash sqrt[3]2\backslash \backslash \; \backslash text\{since\}\backslash \backslash \; \backslash left(\backslash sqrt[3]\{2\}\backslash right)^3=2$

$a^{\frac{1}{n}}=\sqrt[n]{a}a^\{\backslash frac\{1\}\{n\}\}=\backslash sqrt[n]\{a\}$

Unit fractions in the exponent