Exponent rules
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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.
Product Rules
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.
$(1).\,\,\,$ $b^{\displaystyle m} \times b^{\displaystyle n} \,=\, b^{\displaystyle \,m+n}$
$(2).\,\,\,$ $b^{\displaystyle m} \times c^{\displaystyle m} \,=\, (b \times c)^{\displaystyle m}$
Quotient Rules
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.
$(1).\,\,\,$ $\dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} \,=\, b^{\displaystyle \, m-n}$
$(2).\,\,\,$ $\dfrac{b^{\displaystyle m}}{c^{\displaystyle m}} \,=\, \bigg(\dfrac{b}{c}\bigg)^{\displaystyle m}$
Power Rules
In mathematics, there are six types of power rules to find value of a quantity with exponent.
$(1).\,\,\,$ $\big(b^{\displaystyle m}\big)^{\displaystyle n} \,=\, b^{\displaystyle \, m \times n}$
$(2).\,\,\,$ $b^{\displaystyle -m} \,=\, \dfrac{1}{b^{\displaystyle m}}$
$(3).\,\,\,$ $b^{\dfrac{1}{n}} \,=\, \sqrt[\displaystyle n]{b\,\,}$
$(4).\,\,\,$ $b^{\dfrac{m}{n}} \,=\, \sqrt[\displaystyle n]{b^{\displaystyle m}}$
$(5).\,\,\,$ $b^0 \,=\, 1$
$(6).\,\,\,$ $b^1 \,=\, b$
