Laws of Exponents

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.

Product Rules

There are two types of fundamental product laws of exponents.

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

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

Quotient Rules

There are two division rules of indices.

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

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

Power Rules

There are six exponential power rules.

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

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

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

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

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

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