There are three types of properties to study the exponents and here is the list of exponential function rules with proofs in algebraic form for beginners.

There are two types of fundamental product laws of exponents.

$(1)\,\,\,\,\,\,$ $\large b^{\displaystyle m} \times b^{\displaystyle n} = b^{\displaystyle \,m+n}$

$(2)\,\,\,\,\,\,$ $\large b^{\displaystyle m} \times c^{\displaystyle m} = (b \times c)^{\displaystyle m}$

There are two division rules of indices.

$(1)\,\,\,\,\,\,$ $\large \dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} \,=\, b^{\displaystyle m-n}$

$(2)\,\,\,\,\,\,$ $\large \dfrac{b^{\displaystyle m}}{c^{\displaystyle m}} \,=\, {\Bigg(\dfrac{b}{c}\Bigg)}^{\displaystyle m}$

There are six exponential power rules.

$(1)\,\,\,\,\,\,$ $\large {(b^m)}^n \,=\, b^{mn}$

$(2)\,\,\,\,\,\,$ $\large b^{-m} \,=\, \dfrac{1}{b^m}$

$(3)\,\,\,\,\,\,$ $\large b^{\frac{1}{n}} \,=\, \sqrt[\displaystyle n]{b}$

$(4)\,\,\,\,\,\,$ $\large b^{\frac{m}{n}} \,=\, \sqrt[\displaystyle n]{b^m}$

$(5)\,\,\,\,\,\,$ $\large b^0 \,=\, 1$

$(6)\,\,\,\,\,\,$ $\large b^1 \,=\, b$

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