EXPONENTS AND POWERS
Division Property :
The division property of exponents states that when you have a quotient of two exponential expressions with the same base, you can subtract the exponent in the denominator from the exponent in the numerator. Mathematically, it can be expressed as:
Case-I: when m>n
L.H.S = \(\frac{{{a^m}}}{{{a^n}}} = {a^{m - n}}(m > n)\)
\( = a \times a \times a \times ........(m - n\,\,times)\)
\(\frac{{{a^m}}}{{{a^n}}} = {a^{m - n}}(m > n)\)
Ex :- i) \(\frac{{{5^2}}}{{{5^1}}} = {5^{2 - 1}} = 5\)
ii)\(\frac{{{{\left( {\sqrt {\frac{3}{2}} } \right)}^{10}}}}{{{{\left( {\sqrt {\frac{3}{2}} } \right)}^8}}} = {\left( {\sqrt {\frac{3}{2}} } \right)^{10 - 8}} = {\left( {\sqrt {\frac{3}{2}} } \right)^2} = \frac{3}{2}\)
iii)\(\frac{{{{100}^{100}}}}{{{{100}^{100}}}} = {100^{100 - 100}} = {100^0} = 1\)
Note : Any real number having its power ‘o’ is equal to ‘1’
Case-II: when m<n \(\frac{{{a^m}}}{{{a^n}}} = \frac{1}{{{a^{n - m}}}}(m < n)\)
L.H.S \(\frac{{{a^m}}}{{{a^n}}} = \frac{{a \times a \times a \times ........(m\,\,times)}}{{a \times a \times a \times ........(n\,\,times)}}\)
\(= \frac{1}{{a \times a \times a \times ........(n - m\,\,times)}}\)
\(= \frac{1}{{{a^{m - n}}}}\)
\(\therefore \,\,\,\,\,\frac{{{a^m}}}{{{a^n}}} = \frac{1}{{{a^{m - n}}}}\)
Ex :- i) \(\frac{{{{11}^5}}}{{{{11}^{11}}}} = \frac{1}{{{{11}^{11 - 5}}}} = \frac{1}{{{{11}^6}}}\)
ii) \(\frac{{{{\left( {\sqrt 3 } \right)}^{101}}}}{{{{\left( {\sqrt 3 } \right)}^{102}}}} = \frac{1}{{{{\left( {\sqrt 3 } \right)}^{102 - 101}}}} = \frac{1}{{{{\left( {\sqrt 3 } \right)}^1}}} = \frac{1}{{\sqrt 3 }}\)