EXPONENTS AND POWERS
Terms of Exponents (or) laws of Indices
Multiplication Property:
The law of exponents for multiplication states that when you multiply two exponential expressions with the same base, you can add the exponents. Mathematically, it can be expressed as:
\({a^m} \times {a^n} = {a^{m + n}}\)
Here, a is the base, and m and n are the exponents. The rule tells us that when you multiply two exponential expressions with the same base, you can combine them by adding the exponents.
L.H.S \({a^m} \times {a^n} = a \times a \times .........(m\,\,times) \times a \times a \times ........(n\,\,times)\)
\(= a \times a \times a \times .........(m + n\,\,times)\)
\({a^m} \times {a^n} = {a^{m + n}}\)
Ex :- i) \({3^6} \times {3^7} = {3^{6 + 7}} = {3^{13}}\)
ii) \({\left( {\sqrt 2 } \right)^n} \times {\left( {\sqrt 2 } \right)^{13}} = {\left( {\sqrt 2 } \right)^{11 + 13}} = {\left( {\sqrt 2 } \right)^{24}} = {\left( {{2^{\frac{1}{2}}}} \right)^{24}} = {2^{12}}\)
iii) \({\left( {\frac{3}{2}} \right)^{\frac{5}{2}}} \times {\left( {\frac{3}{2}} \right)^{\frac{2}{5}}} = {\left( {\frac{3}{2}} \right)^{\frac{5}{2} + \frac{2}{5}}} = {\left( {\frac{3}{2}} \right)^{\frac{{29}}{{10}}}}\)
NOTE: This rule is applicable as long as the bases are the same. If the bases are different, you cannot use this multiplication property directly