| Category | Law | Examples | Comments |
|---|---|---|---|
| Multiplication | \(a^xa^y = a^{a+y}\) | Example 1: \[\begin{aligned} 2^3\times2^5\\ =(2\times2\times2)\times(2\times2\times2\times2\times2)\\ =2^8 \end{aligned}\] Example 2: \[\begin{aligned} 3^4\times3^2\times3\\ =3^4\times3^2\times3^1\\ =3^{4+2+1}\\ =3^7 \end{aligned}\] Example 3: \[\begin{aligned} 6^3\times15^2\times10^5\\ =(2^3\times3^3)\times(3^2\times5^2)\times(5^5\times2^5)\\ =2^8\times3^5\times5^7 \end{aligned}\] |
When you multiply by powers of 2 you add the indices. |
| Division | \(\frac{a^x}{a^y}=a^{x-y}\) | Example 1: \[\begin{aligned} \frac{2^8}{2^5}\\ =\frac{2\times2\times2\times2\times2\times2\times2\times2}{2\times2\times2\times2\times2}\\ =\frac{2}{2}\times\frac{2}{2}\times\frac{2}{2}\times\frac{2}{2}\times\frac{2}{2}\times\frac{2}{1}\times\frac{2}{1}\times\frac{2}{1}\\ =1\times1\times1\times1\times1\times2\times2\times2\\ =2^3 \end{aligned}\] Example 2: \[\begin{aligned} \frac{15^3\times3^2}{5^2}\\ =\frac{3^3\times5^3\times3^2}{5^2}\\ =3^{3+2}\times5^{3-2}\\ =3^5\times5 \end{aligned}\] |
When you divide by the powers of 2, you subtract the indices. |
| Powers | \((a^x)^y=a^{xy}\) | Example 1: \[\begin{aligned} (3^5)^4\\ =3^5\times3^5\times3^5\times3^5\\ =3^{20} \end{aligned}\] |
When you raise a power by another power you multiply the indices. |
| Other rules | \(a^0=1\) (for \(a>0\)) \(a^{-x}=\frac{1}{a^x}\) \(a^{\frac{y}{x}}=\sqrt[x]{a^y}\) |