The quantities are expressed in exponential notation in mathematics for some reasons. For doing some mathematical activities with exponential functions, the fundamental formulas of mathematics may not be useful. Hence, it requires some special properties and they are called the rules of exponents. Now, let’s learn the following three types of exponential properties with proofs to study the indices (or exponents) mathematically.

There are two types of product laws in mathematics and they are used to multiply a quantity in exponential form by another quantity in exponential notation.

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

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

There are two types of division laws in mathematics, they are used to divide a quantity in exponential form by another quantity in exponential form.

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

$(2).\,\,\,$ $\dfrac{b^{\displaystyle m}}{c^{\displaystyle m}} \,=\, \bigg(\dfrac{b}{c}\bigg)^{\displaystyle m}$

In mathematics, there are six types of power rules to find value of a quantity with exponent.

$(1).\,\,\,$ $\big(b^{\displaystyle m}\big)^{\displaystyle n} \,=\, b^{\displaystyle \, m \times n}$

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

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

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

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

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

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